| L(s) = 1 | − 1.71·2-s + 0.260·3-s + 0.927·4-s − 0.445·6-s − 7-s + 1.83·8-s − 2.93·9-s + 2.25·11-s + 0.241·12-s + 13-s + 1.71·14-s − 4.99·16-s − 0.200·17-s + 5.01·18-s − 4.90·19-s − 0.260·21-s − 3.85·22-s + 7.80·23-s + 0.477·24-s − 1.71·26-s − 1.54·27-s − 0.927·28-s − 1.52·29-s + 3.40·31-s + 4.87·32-s + 0.586·33-s + 0.342·34-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.150·3-s + 0.463·4-s − 0.181·6-s − 0.377·7-s + 0.649·8-s − 0.977·9-s + 0.679·11-s + 0.0696·12-s + 0.277·13-s + 0.457·14-s − 1.24·16-s − 0.0485·17-s + 1.18·18-s − 1.12·19-s − 0.0568·21-s − 0.821·22-s + 1.62·23-s + 0.0975·24-s − 0.335·26-s − 0.297·27-s − 0.175·28-s − 0.284·29-s + 0.612·31-s + 0.861·32-s + 0.102·33-s + 0.0587·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 - 0.260T + 3T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 17 | \( 1 + 0.200T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 - 7.80T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 37 | \( 1 + 0.981T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 7.66T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 - 8.78T + 61T^{2} \) |
| 67 | \( 1 + 0.421T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 0.220T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 5.34T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.764268592239878721678563449751, −8.243908538489385966361653342603, −7.12975178707450399184267290464, −6.62987990702188135492968477295, −5.60222816619267163876346071959, −4.58237862371352855916400724246, −3.55408255664486375514320732909, −2.49535443681621444533367883576, −1.29550729632649451064626672753, 0,
1.29550729632649451064626672753, 2.49535443681621444533367883576, 3.55408255664486375514320732909, 4.58237862371352855916400724246, 5.60222816619267163876346071959, 6.62987990702188135492968477295, 7.12975178707450399184267290464, 8.243908538489385966361653342603, 8.764268592239878721678563449751
