| L(s) = 1 | + 1.81·2-s + 3.10·3-s + 1.28·4-s + 5.62·6-s + 7-s − 1.28·8-s + 6.62·9-s + 3.10·11-s + 3.99·12-s − 13-s + 1.81·14-s − 4.91·16-s + 0.524·17-s + 12.0·18-s + 0.813·19-s + 3.10·21-s + 5.62·22-s − 7.33·23-s − 4.00·24-s − 1.81·26-s + 11.2·27-s + 1.28·28-s + 8.28·29-s + 1.39·31-s − 6.33·32-s + 9.62·33-s + 0.951·34-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 1.79·3-s + 0.644·4-s + 2.29·6-s + 0.377·7-s − 0.455·8-s + 2.20·9-s + 0.935·11-s + 1.15·12-s − 0.277·13-s + 0.484·14-s − 1.22·16-s + 0.127·17-s + 2.83·18-s + 0.186·19-s + 0.677·21-s + 1.19·22-s − 1.53·23-s − 0.816·24-s − 0.355·26-s + 2.16·27-s + 0.243·28-s + 1.53·29-s + 0.250·31-s − 1.12·32-s + 1.67·33-s + 0.163·34-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
| \(L(1)\) |
\(\approx\) |
\(6.613895247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.613895247\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.81T + 2T^{2} \) |
| 3 | \( 1 - 3.10T + 3T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 17 | \( 1 - 0.524T + 17T^{2} \) |
| 19 | \( 1 - 0.813T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 - 2.49T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 8.72T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 1.18T + 97T^{2} \) |
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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
−8.823984787396605357701329269941, −8.363985202029440328392747742534, −7.49147077640232604228675064854, −6.66939625279353381482979186062, −5.80317565623292017931052684358, −4.51574009253388545405554850548, −4.21015316889081536562101511777, −3.27675977183679572257693561225, −2.61212740189944162147350397224, −1.58119057474533452137850714711,
1.58119057474533452137850714711, 2.61212740189944162147350397224, 3.27675977183679572257693561225, 4.21015316889081536562101511777, 4.51574009253388545405554850548, 5.80317565623292017931052684358, 6.66939625279353381482979186062, 7.49147077640232604228675064854, 8.363985202029440328392747742534, 8.823984787396605357701329269941
