| L(s) = 1 | − 22·9-s − 16·29-s + 58·49-s − 32·61-s − 56·79-s + 241·81-s + 64·101-s + 92·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 7.33·9-s − 2.97·29-s + 58/7·49-s − 4.09·61-s − 6.30·79-s + 26.7·81-s + 6.36·101-s + 8.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.116033739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.116033739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 18 T^{2} + 309 T^{4} + 5528 T^{6} + 80242 T^{8} + 1027820 T^{10} + 80242 p^{2} T^{12} + 5528 p^{4} T^{14} + 309 p^{6} T^{16} + 18 p^{8} T^{18} + p^{10} T^{20} \) |
| good | 3 | \( ( 1 + 11 T^{2} + 61 T^{4} + 212 T^{6} + 554 T^{8} + 1442 T^{10} + 554 p^{2} T^{12} + 212 p^{4} T^{14} + 61 p^{6} T^{16} + 11 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 7 | \( ( 1 - 29 T^{2} + 67 p T^{4} - 5484 T^{6} + 51994 T^{8} - 402478 T^{10} + 51994 p^{2} T^{12} - 5484 p^{4} T^{14} + 67 p^{7} T^{16} - 29 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 11 | \( ( 1 - 46 T^{2} + 965 T^{4} - 13192 T^{6} + 147954 T^{8} - 1605460 T^{10} + 147954 p^{2} T^{12} - 13192 p^{4} T^{14} + 965 p^{6} T^{16} - 46 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 17 | \( ( 1 + 63 T^{2} + 2517 T^{4} + 72708 T^{6} + 1654922 T^{8} + 31028090 T^{10} + 1654922 p^{2} T^{12} + 72708 p^{4} T^{14} + 2517 p^{6} T^{16} + 63 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 19 | \( ( 1 - 98 T^{2} + 4949 T^{4} - 175160 T^{6} + 4736210 T^{8} - 100809868 T^{10} + 4736210 p^{2} T^{12} - 175160 p^{4} T^{14} + 4949 p^{6} T^{16} - 98 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 23 | \( ( 1 + 186 T^{2} + 16317 T^{4} + 889304 T^{6} + 33346578 T^{8} + 898888412 T^{10} + 33346578 p^{2} T^{12} + 889304 p^{4} T^{14} + 16317 p^{6} T^{16} + 186 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 29 | \( ( 1 + 4 T + 89 T^{2} + 144 T^{3} + 3234 T^{4} + 1688 T^{5} + 3234 p T^{6} + 144 p^{2} T^{7} + 89 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 31 | \( ( 1 - 202 T^{2} + 18957 T^{4} - 1128280 T^{6} + 49146482 T^{8} - 1692179260 T^{10} + 49146482 p^{2} T^{12} - 1128280 p^{4} T^{14} + 18957 p^{6} T^{16} - 202 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 37 | \( ( 1 - 305 T^{2} + 43597 T^{4} - 3851740 T^{6} + 233248906 T^{8} - 10126442054 T^{10} + 233248906 p^{2} T^{12} - 3851740 p^{4} T^{14} + 43597 p^{6} T^{16} - 305 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 41 | \( ( 1 - 202 T^{2} + 21245 T^{4} - 1520632 T^{6} + 83484562 T^{8} - 3747733244 T^{10} + 83484562 p^{2} T^{12} - 1520632 p^{4} T^{14} + 21245 p^{6} T^{16} - 202 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 43 | \( ( 1 + 219 T^{2} + 26861 T^{4} + 2252564 T^{6} + 141711914 T^{8} + 6884853698 T^{10} + 141711914 p^{2} T^{12} + 2252564 p^{4} T^{14} + 26861 p^{6} T^{16} + 219 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 47 | \( ( 1 - 429 T^{2} + 84549 T^{4} - 10077804 T^{6} + 804363674 T^{8} - 44914628558 T^{10} + 804363674 p^{2} T^{12} - 10077804 p^{4} T^{14} + 84549 p^{6} T^{16} - 429 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 53 | \( ( 1 + 166 T^{2} + 19301 T^{4} + 1717192 T^{6} + 122805394 T^{8} + 7039697252 T^{10} + 122805394 p^{2} T^{12} + 1717192 p^{4} T^{14} + 19301 p^{6} T^{16} + 166 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 59 | \( ( 1 - 274 T^{2} + 40005 T^{4} - 4124152 T^{6} + 333328658 T^{8} - 21783115564 T^{10} + 333328658 p^{2} T^{12} - 4124152 p^{4} T^{14} + 40005 p^{6} T^{16} - 274 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 61 | \( ( 1 + 8 T + 185 T^{2} + 1280 T^{3} + 14562 T^{4} + 96752 T^{5} + 14562 p T^{6} + 1280 p^{2} T^{7} + 185 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 67 | \( ( 1 - 446 T^{2} + 101269 T^{4} - 14909736 T^{6} + 1562377714 T^{8} - 121030224628 T^{10} + 1562377714 p^{2} T^{12} - 14909736 p^{4} T^{14} + 101269 p^{6} T^{16} - 446 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 71 | \( ( 1 - 235 T^{2} + 27925 T^{4} - 2415092 T^{6} + 207629498 T^{8} - 16205410050 T^{10} + 207629498 p^{2} T^{12} - 2415092 p^{4} T^{14} + 27925 p^{6} T^{16} - 235 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 73 | \( ( 1 - 518 T^{2} + 131389 T^{4} - 21282760 T^{6} + 2435057554 T^{8} - 205591832036 T^{10} + 2435057554 p^{2} T^{12} - 21282760 p^{4} T^{14} + 131389 p^{6} T^{16} - 518 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 79 | \( ( 1 + 14 T + 251 T^{2} + 2248 T^{3} + 33402 T^{4} + 264404 T^{5} + 33402 p T^{6} + 2248 p^{2} T^{7} + 251 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 83 | \( ( 1 - 202 T^{2} + 41685 T^{4} - 4982904 T^{6} + 603379730 T^{8} - 50262539964 T^{10} + 603379730 p^{2} T^{12} - 4982904 p^{4} T^{14} + 41685 p^{6} T^{16} - 202 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 89 | \( ( 1 - 366 T^{2} + 71421 T^{4} - 10162664 T^{6} + 1185209586 T^{8} - 115893586388 T^{10} + 1185209586 p^{2} T^{12} - 10162664 p^{4} T^{14} + 71421 p^{6} T^{16} - 366 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 97 | \( ( 1 - 506 T^{2} + 123597 T^{4} - 19841016 T^{6} + 2422744562 T^{8} - 249812113948 T^{10} + 2422744562 p^{2} T^{12} - 19841016 p^{4} T^{14} + 123597 p^{6} T^{16} - 506 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.91483728680896960867536620214, −1.78893856720956338228675344294, −1.78857750638616321217703597728, −1.72806248524795131767953981520, −1.70697597963074830289554617516, −1.60015709880850443246246656042, −1.59140019573609181314951928363, −1.37095892816697631008352488244, −1.22260873030949707599208140661, −1.15833828555616365341920097904, −1.15112551905321348959499820495, −1.09350528346968520262217074688, −1.07058536607244892028825344365, −0.950782544221473328299216402275, −0.929836205596706707684490832348, −0.813810932837885060900259491399, −0.66412345455531825608589707812, −0.58896763604187576819040823859, −0.54311624749410560118550733797, −0.48840768749462929875708273366, −0.41763274897209169580123528223, −0.35175337323089171146106470239, −0.27186865707895990728374793173, −0.17687439609404559884057875157, −0.12155337158127639282056821841,
0.12155337158127639282056821841, 0.17687439609404559884057875157, 0.27186865707895990728374793173, 0.35175337323089171146106470239, 0.41763274897209169580123528223, 0.48840768749462929875708273366, 0.54311624749410560118550733797, 0.58896763604187576819040823859, 0.66412345455531825608589707812, 0.813810932837885060900259491399, 0.929836205596706707684490832348, 0.950782544221473328299216402275, 1.07058536607244892028825344365, 1.09350528346968520262217074688, 1.15112551905321348959499820495, 1.15833828555616365341920097904, 1.22260873030949707599208140661, 1.37095892816697631008352488244, 1.59140019573609181314951928363, 1.60015709880850443246246656042, 1.70697597963074830289554617516, 1.72806248524795131767953981520, 1.78857750638616321217703597728, 1.78893856720956338228675344294, 1.91483728680896960867536620214
Plot not available for L-functions of degree greater than 10.
