| L(s) = 1 | − 2.51·2-s + 4.32·4-s + 5-s + 3.32·7-s − 5.83·8-s − 2.51·10-s + 2.83·11-s − 8.34·14-s + 6.02·16-s + 6.64·17-s + 2.19·19-s + 4.32·20-s − 7.12·22-s + 0.485·23-s + 25-s + 14.3·28-s + 3.32·29-s + 3.80·31-s − 3.48·32-s − 16.6·34-s + 3.32·35-s + 9.32·37-s − 5.51·38-s − 5.83·40-s + 1.61·41-s − 0.872·43-s + 12.2·44-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.16·4-s + 0.447·5-s + 1.25·7-s − 2.06·8-s − 0.795·10-s + 0.854·11-s − 2.23·14-s + 1.50·16-s + 1.61·17-s + 0.503·19-s + 0.966·20-s − 1.51·22-s + 0.101·23-s + 0.200·25-s + 2.71·28-s + 0.616·29-s + 0.683·31-s − 0.616·32-s − 2.86·34-s + 0.561·35-s + 1.53·37-s − 0.894·38-s − 0.922·40-s + 0.251·41-s − 0.133·43-s + 1.84·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516451997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516451997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 - 2.83T + 11T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 - 0.485T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 3.80T + 31T^{2} \) |
| 37 | \( 1 - 9.32T + 37T^{2} \) |
| 41 | \( 1 - 1.61T + 41T^{2} \) |
| 43 | \( 1 + 0.872T + 43T^{2} \) |
| 47 | \( 1 - 3.32T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.83T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 4.29T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 0.585T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 - 0.641T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.048981790741261376866248836078, −7.51004515751975314543270252490, −6.73171812368204319204339380469, −6.04294500494271964389574068991, −5.25887338773235956524320414489, −4.34945047894899629539736310235, −3.16680361381545679195542048337, −2.28697515783435572310225539325, −1.32652502207087293819551576700, −0.979648655266459254706469274800,
0.979648655266459254706469274800, 1.32652502207087293819551576700, 2.28697515783435572310225539325, 3.16680361381545679195542048337, 4.34945047894899629539736310235, 5.25887338773235956524320414489, 6.04294500494271964389574068991, 6.73171812368204319204339380469, 7.51004515751975314543270252490, 8.048981790741261376866248836078
