| L(s) = 1 | − 2-s + 4-s − 3.77·5-s − 0.643·7-s − 8-s + 3.77·10-s + 5.83·11-s + 5.33·13-s + 0.643·14-s + 16-s + 2.05·17-s − 5.18·19-s − 3.77·20-s − 5.83·22-s + 9.24·25-s − 5.33·26-s − 0.643·28-s − 8.42·29-s − 8.24·31-s − 32-s − 2.05·34-s + 2.42·35-s + 0.302·37-s + 5.18·38-s + 3.77·40-s − 7.58·41-s + 9.90·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.68·5-s − 0.243·7-s − 0.353·8-s + 1.19·10-s + 1.75·11-s + 1.48·13-s + 0.171·14-s + 0.250·16-s + 0.499·17-s − 1.19·19-s − 0.844·20-s − 1.24·22-s + 1.84·25-s − 1.04·26-s − 0.121·28-s − 1.56·29-s − 1.48·31-s − 0.176·32-s − 0.352·34-s + 0.410·35-s + 0.0497·37-s + 0.841·38-s + 0.596·40-s − 1.18·41-s + 1.51·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 + 0.643T + 7T^{2} \) |
| 11 | \( 1 - 5.83T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 + 8.24T + 31T^{2} \) |
| 37 | \( 1 - 0.302T + 37T^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 - 9.90T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 + 0.340T + 53T^{2} \) |
| 59 | \( 1 + 5.81T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 6.12T + 67T^{2} \) |
| 71 | \( 1 + 3.09T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−7.49366356628794747912867891273, −6.70325791659145882626379391083, −6.35772308067818466292108916668, −5.39010514389655976792543517274, −4.14356795387941161171026184086, −3.78006390616766590615065440676, −3.35955968083481532695654885901, −1.89246103781075669059935517853, −1.05054128298984746796944404561, 0,
1.05054128298984746796944404561, 1.89246103781075669059935517853, 3.35955968083481532695654885901, 3.78006390616766590615065440676, 4.14356795387941161171026184086, 5.39010514389655976792543517274, 6.35772308067818466292108916668, 6.70325791659145882626379391083, 7.49366356628794747912867891273
