| L(s) = 1 | − 3·3-s − 2·4-s + 2·5-s − 4·7-s + 6·9-s + 2·11-s + 6·12-s + 6·13-s − 6·15-s + 4·16-s + 19-s − 4·20-s + 12·21-s − 25-s − 9·27-s + 8·28-s + 9·29-s + 9·31-s − 6·33-s − 8·35-s − 12·36-s − 2·37-s − 18·39-s + 6·41-s − 43-s − 4·44-s + 12·45-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s + 0.894·5-s − 1.51·7-s + 2·9-s + 0.603·11-s + 1.73·12-s + 1.66·13-s − 1.54·15-s + 16-s + 0.229·19-s − 0.894·20-s + 2.61·21-s − 1/5·25-s − 1.73·27-s + 1.51·28-s + 1.67·29-s + 1.61·31-s − 1.04·33-s − 1.35·35-s − 2·36-s − 0.328·37-s − 2.88·39-s + 0.937·41-s − 0.152·43-s − 0.603·44-s + 1.78·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5491 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5491 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9051889917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9051889917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.348822695521017730103455379704, −7.03356536705576559773788368531, −6.32584185844515214531158257977, −6.08277479954650942845427293748, −5.53075951394213784356181927648, −4.56740650526253621802725397288, −3.94588112671431559655876248649, −3.00555111549708386532792922576, −1.30513755513049493467280969242, −0.65549664568417250284040685903,
0.65549664568417250284040685903, 1.30513755513049493467280969242, 3.00555111549708386532792922576, 3.94588112671431559655876248649, 4.56740650526253621802725397288, 5.53075951394213784356181927648, 6.08277479954650942845427293748, 6.32584185844515214531158257977, 7.03356536705576559773788368531, 8.348822695521017730103455379704
