| L(s) = 1 | + 1.19·2-s − 3.16·3-s − 0.576·4-s − 3.77·6-s + 1.30·7-s − 3.07·8-s + 7.00·9-s − 11-s + 1.82·12-s − 2.53·13-s + 1.56·14-s − 2.51·16-s + 5.43·17-s + 8.35·18-s + 19-s − 4.14·21-s − 1.19·22-s − 3.87·23-s + 9.72·24-s − 3.03·26-s − 12.6·27-s − 0.754·28-s − 2.41·29-s + 3.03·31-s + 3.14·32-s + 3.16·33-s + 6.48·34-s + ⋯ |
| L(s) = 1 | + 0.843·2-s − 1.82·3-s − 0.288·4-s − 1.54·6-s + 0.494·7-s − 1.08·8-s + 2.33·9-s − 0.301·11-s + 0.526·12-s − 0.704·13-s + 0.417·14-s − 0.628·16-s + 1.31·17-s + 1.96·18-s + 0.229·19-s − 0.903·21-s − 0.254·22-s − 0.807·23-s + 1.98·24-s − 0.594·26-s − 2.43·27-s − 0.142·28-s − 0.448·29-s + 0.545·31-s + 0.556·32-s + 0.550·33-s + 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9236923487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9236923487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 3 | \( 1 + 3.16T + 3T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 6.11T + 41T^{2} \) |
| 43 | \( 1 - 2.95T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 0.992T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 + 5.84T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 - 7.01T + 83T^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.891108858679997076319501187587, −7.35319509215355337987132999713, −6.38777247523949585878708514680, −5.80090171755709539531128738765, −5.26359517317223457558953673447, −4.78009566681599537957305680344, −4.09190014295302490394205399657, −3.12918528060248249361018131039, −1.70246498799786798852036310025, −0.51120619779012632961477916165,
0.51120619779012632961477916165, 1.70246498799786798852036310025, 3.12918528060248249361018131039, 4.09190014295302490394205399657, 4.78009566681599537957305680344, 5.26359517317223457558953673447, 5.80090171755709539531128738765, 6.38777247523949585878708514680, 7.35319509215355337987132999713, 7.891108858679997076319501187587
