| L(s) = 1 | − 7·16-s + 32·31-s − 8·61-s + 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 7/4·16-s + 5.74·31-s − 1.02·61-s + 8·121-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 + 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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.734989199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.734989199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 7 T^{4} + 33 T^{8} + 7 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 17 | \( 1 + 382 T^{4} + 62403 T^{8} + 382 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 22 T^{2} + 123 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 98 T^{4} - 270237 T^{8} - 98 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 4222 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 1778 T^{4} - 4729197 T^{8} - 1778 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 98 T^{2} + 3363 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 9938 T^{4} + 51305523 T^{8} - 9938 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.50235000422917169744588407396, −4.45762626248600924569036975361, −4.44905807308022486202030462283, −4.37131372554423831662910068585, −4.23064595407003368589201387039, −3.73831635165887716450178580817, −3.71462654458107191095121695518, −3.55633691482222413475056151762, −3.47889511889981401177995152310, −3.32178935564624347389252132944, −3.07561148599495689168122107459, −2.96061137310652644540671108026, −2.82816656527892133189231335969, −2.65657196839157246553384707859, −2.46986461457288465395375231470, −2.39294481112280531745421169163, −2.23498214064593349484839639990, −2.00104966607839575019946129592, −1.91731487361627596950937507301, −1.60122855592408167523129683536, −1.18268036343089796183575161978, −1.10195079868744653499391282298, −1.04948987970721378391316480596, −0.60981317865546148176813637101, −0.26603576603710216863572193137,
0.26603576603710216863572193137, 0.60981317865546148176813637101, 1.04948987970721378391316480596, 1.10195079868744653499391282298, 1.18268036343089796183575161978, 1.60122855592408167523129683536, 1.91731487361627596950937507301, 2.00104966607839575019946129592, 2.23498214064593349484839639990, 2.39294481112280531745421169163, 2.46986461457288465395375231470, 2.65657196839157246553384707859, 2.82816656527892133189231335969, 2.96061137310652644540671108026, 3.07561148599495689168122107459, 3.32178935564624347389252132944, 3.47889511889981401177995152310, 3.55633691482222413475056151762, 3.71462654458107191095121695518, 3.73831635165887716450178580817, 4.23064595407003368589201387039, 4.37131372554423831662910068585, 4.44905807308022486202030462283, 4.45762626248600924569036975361, 4.50235000422917169744588407396
Plot not available for L-functions of degree greater than 10.
