| L(s) = 1 | + 1.81·2-s + 1.28·4-s + 2.81·5-s − 1.28·8-s + 5.10·10-s − 3.10·11-s − 13-s − 4.91·16-s − 0.524·17-s − 0.813·19-s + 3.62·20-s − 5.62·22-s − 7.33·23-s + 2.91·25-s − 1.81·26-s − 8.28·29-s − 1.39·31-s − 6.33·32-s − 0.951·34-s − 6.15·37-s − 1.47·38-s − 3.62·40-s − 4.20·41-s + 6.75·43-s − 3.99·44-s − 13.3·46-s − 5.97·47-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.644·4-s + 1.25·5-s − 0.455·8-s + 1.61·10-s − 0.935·11-s − 0.277·13-s − 1.22·16-s − 0.127·17-s − 0.186·19-s + 0.811·20-s − 1.19·22-s − 1.53·23-s + 0.583·25-s − 0.355·26-s − 1.53·29-s − 0.250·31-s − 1.12·32-s − 0.163·34-s − 1.01·37-s − 0.239·38-s − 0.573·40-s − 0.656·41-s + 1.03·43-s − 0.603·44-s − 1.96·46-s − 0.870·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 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 | \( 1 - 1.18T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61929158389835206292358108821, −6.73557186223943787482678231608, −6.04506516239010667841791313432, −5.49495758383364889825744542396, −5.06898064119564441732956838655, −4.11710879585163276589862533402, −3.37365015269456516225678205651, −2.35769717024745362005061856055, −1.91024563837143301616346394150, 0,
1.91024563837143301616346394150, 2.35769717024745362005061856055, 3.37365015269456516225678205651, 4.11710879585163276589862533402, 5.06898064119564441732956838655, 5.49495758383364889825744542396, 6.04506516239010667841791313432, 6.73557186223943787482678231608, 7.61929158389835206292358108821
