| L(s) = 1 | + 458·9-s + 400·11-s − 1.68e3·19-s + 9.36e3·29-s + 1.00e4·31-s − 1.06e4·41-s + 6.52e4·49-s − 1.63e5·59-s − 9.38e4·61-s − 1.48e4·71-s − 2.16e5·79-s + 5.46e4·81-s − 1.41e5·89-s + 1.83e5·99-s − 3.02e5·101-s − 2.82e5·109-s − 3.31e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.61e3·169-s + ⋯ |
| L(s) = 1 | + 1.88·9-s + 0.996·11-s − 1.06·19-s + 2.06·29-s + 1.87·31-s − 0.991·41-s + 3.88·49-s − 6.11·59-s − 3.22·61-s − 0.350·71-s − 3.89·79-s + 0.925·81-s − 1.89·89-s + 1.87·99-s − 2.94·101-s − 2.28·109-s − 2.05·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.0178·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.107307668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107307668\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 458 T^{2} + 1915 p^{4} T^{4} - 458 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1332 p^{2} T^{2} + 1629866758 T^{4} - 1332 p^{12} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 200 T + 225821 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6612 T^{2} + 47343141238 T^{4} - 6612 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5517394 T^{2} + 11637628825043 T^{4} - 5517394 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 840 T + 4289677 T^{2} + 840 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 19211092 T^{2} + 166300343917958 T^{4} - 19211092 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4680 T + 1421346 p T^{2} - 4680 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 5008 T + 33327162 T^{2} - 5008 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 86339436 T^{2} + 2659590797035222 T^{4} - 86339436 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 5334 T + 27686155 T^{2} + 5334 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 29960652 T^{2} + 43433374484607958 T^{4} - 29960652 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 288752718 T^{2} + p^{10} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 514633100 T^{2} + 347542581493770198 T^{4} - 514633100 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 81776 T + 3059970646 T^{2} + 81776 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 46932 T + 2239379182 T^{2} + 46932 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2982943418 T^{2} + 5751121538625041883 T^{4} - 2982943418 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 7448 T + 3593807902 T^{2} + 7448 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 2368673842 T^{2} + 9983110585747236243 T^{4} - 2368673842 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 108104 T + 6816153098 T^{2} + 108104 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 8274508042 T^{2} + 45513777799710942443 T^{4} - 8274508042 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 70990 T + 6345035107 T^{2} + 70990 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 22699126076 T^{2} + \)\(24\!\cdots\!42\)\( T^{4} - 22699126076 p^{10} T^{6} + p^{20} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.22171930189438933714770280347, −7.18149782651507435171991932914, −6.92056930323724152326141421754, −6.35190325457508689774859263014, −6.24870478256814141059083090402, −6.24655831961947912347772023758, −6.11243150780367246229444345206, −5.51126758930304394086993144781, −5.13284249925628783845851612813, −4.94753973206135619990763114506, −4.57507322350901849406156385268, −4.28639793002614092324151207971, −4.27634963470377505033110473711, −3.96797720615073122359533515122, −3.89774485908660560905560629565, −2.99789916525418945741066963265, −2.92925878728820146068957485401, −2.72809883092505504913273257748, −2.54182571481102836277657297841, −1.64551414249457844587245147104, −1.39210243294453318986926915077, −1.36512564553566985843762672332, −1.36185935200599537084277574108, −0.57078583167653498487124373135, −0.11960983469108833787289390937,
0.11960983469108833787289390937, 0.57078583167653498487124373135, 1.36185935200599537084277574108, 1.36512564553566985843762672332, 1.39210243294453318986926915077, 1.64551414249457844587245147104, 2.54182571481102836277657297841, 2.72809883092505504913273257748, 2.92925878728820146068957485401, 2.99789916525418945741066963265, 3.89774485908660560905560629565, 3.96797720615073122359533515122, 4.27634963470377505033110473711, 4.28639793002614092324151207971, 4.57507322350901849406156385268, 4.94753973206135619990763114506, 5.13284249925628783845851612813, 5.51126758930304394086993144781, 6.11243150780367246229444345206, 6.24655831961947912347772023758, 6.24870478256814141059083090402, 6.35190325457508689774859263014, 6.92056930323724152326141421754, 7.18149782651507435171991932914, 7.22171930189438933714770280347
