| L(s) = 1 | − 2-s + 4-s + 3.20·5-s + 1.51·7-s − 8-s − 3.20·10-s + 6.53·11-s + 2.88·13-s − 1.51·14-s + 16-s + 1.19·17-s + 0.555·19-s + 3.20·20-s − 6.53·22-s + 5.26·25-s − 2.88·26-s + 1.51·28-s + 8.66·29-s − 6.73·31-s − 32-s − 1.19·34-s + 4.84·35-s − 1.45·37-s − 0.555·38-s − 3.20·40-s + 1.84·41-s + 2.17·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.43·5-s + 0.571·7-s − 0.353·8-s − 1.01·10-s + 1.97·11-s + 0.800·13-s − 0.404·14-s + 0.250·16-s + 0.290·17-s + 0.127·19-s + 0.716·20-s − 1.39·22-s + 1.05·25-s − 0.566·26-s + 0.285·28-s + 1.60·29-s − 1.20·31-s − 0.176·32-s − 0.205·34-s + 0.819·35-s − 0.239·37-s − 0.0901·38-s − 0.506·40-s + 0.288·41-s + 0.331·43-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
| \(L(1)\) |
\(\approx\) |
\(2.980772842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.980772842\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.20T + 5T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 - 6.53T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 1.19T + 17T^{2} \) |
| 19 | \( 1 - 0.555T + 19T^{2} \) |
| 29 | \( 1 - 8.66T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 + 1.45T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 - 9.03T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 - 9.45T + 71T^{2} \) |
| 73 | \( 1 + 0.627T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 + 7.24T + 97T^{2} \) |
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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
−7.75383751608546160684497109250, −6.89412770527393813910367311763, −6.33749461154068230578908822409, −5.92247020959807950258667972694, −5.08785239377675171720074873745, −4.15568572030804847072048245795, −3.33689430731299423605738325138, −2.29291715782883045782446512496, −1.48733976333243182442186602372, −1.08274540701037594419077880527,
1.08274540701037594419077880527, 1.48733976333243182442186602372, 2.29291715782883045782446512496, 3.33689430731299423605738325138, 4.15568572030804847072048245795, 5.08785239377675171720074873745, 5.92247020959807950258667972694, 6.33749461154068230578908822409, 6.89412770527393813910367311763, 7.75383751608546160684497109250
