| L(s) = 1 | + 0.679·2-s − 2.20·3-s − 1.53·4-s − 1.49·6-s + 5.10·7-s − 2.40·8-s + 1.84·9-s + 3.02·11-s + 3.38·12-s − 0.110·13-s + 3.47·14-s + 1.44·16-s + 0.309·17-s + 1.25·18-s + 4.32·19-s − 11.2·21-s + 2.05·22-s + 3.82·23-s + 5.29·24-s − 0.0748·26-s + 2.54·27-s − 7.85·28-s − 5.39·29-s + 8.39·31-s + 5.78·32-s − 6.65·33-s + 0.210·34-s + ⋯ |
| L(s) = 1 | + 0.480·2-s − 1.27·3-s − 0.769·4-s − 0.610·6-s + 1.93·7-s − 0.850·8-s + 0.613·9-s + 0.912·11-s + 0.977·12-s − 0.0305·13-s + 0.927·14-s + 0.360·16-s + 0.0751·17-s + 0.294·18-s + 0.992·19-s − 2.45·21-s + 0.438·22-s + 0.798·23-s + 1.07·24-s − 0.0146·26-s + 0.490·27-s − 1.48·28-s − 1.00·29-s + 1.50·31-s + 1.02·32-s − 1.15·33-s + 0.0360·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.824216046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824216046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.679T + 2T^{2} \) |
| 3 | \( 1 + 2.20T + 3T^{2} \) |
| 7 | \( 1 - 5.10T + 7T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 13 | \( 1 + 0.110T + 13T^{2} \) |
| 17 | \( 1 - 0.309T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 + 5.39T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 6.25T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 + 4.42T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 + 5.08T + 71T^{2} \) |
| 73 | \( 1 - 4.97T + 73T^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 + 3.89T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 8.67T + 97T^{2} \) |
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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
−8.049550297428958871674841992839, −7.34737550126127625537032185781, −6.38543670091983102483541995974, −5.71180658598004791082666269411, −5.12107414751724645867777957352, −4.65066399422121372794523458024, −4.07932777667469402715594699405, −2.92624956431124298817658604499, −1.47548522187730492252556411234, −0.804147570054730357468171607047,
0.804147570054730357468171607047, 1.47548522187730492252556411234, 2.92624956431124298817658604499, 4.07932777667469402715594699405, 4.65066399422121372794523458024, 5.12107414751724645867777957352, 5.71180658598004791082666269411, 6.38543670091983102483541995974, 7.34737550126127625537032185781, 8.049550297428958871674841992839
