| L(s) = 1 | + 0.549·2-s + 3-s − 1.69·4-s + 1.27·5-s + 0.549·6-s − 2.63·7-s − 2.03·8-s + 9-s + 0.702·10-s + 1.22·11-s − 1.69·12-s − 13-s − 1.44·14-s + 1.27·15-s + 2.28·16-s + 7.62·17-s + 0.549·18-s − 3.87·19-s − 2.17·20-s − 2.63·21-s + 0.672·22-s − 3.23·23-s − 2.03·24-s − 3.36·25-s − 0.549·26-s + 27-s + 4.48·28-s + ⋯ |
| L(s) = 1 | + 0.388·2-s + 0.577·3-s − 0.849·4-s + 0.571·5-s + 0.224·6-s − 0.997·7-s − 0.718·8-s + 0.333·9-s + 0.222·10-s + 0.369·11-s − 0.490·12-s − 0.277·13-s − 0.387·14-s + 0.330·15-s + 0.570·16-s + 1.85·17-s + 0.129·18-s − 0.889·19-s − 0.485·20-s − 0.575·21-s + 0.143·22-s − 0.674·23-s − 0.414·24-s − 0.672·25-s − 0.107·26-s + 0.192·27-s + 0.846·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.549T + 2T^{2} \) |
| 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 + 2.63T + 7T^{2} \) |
| 11 | \( 1 - 1.22T + 11T^{2} \) |
| 17 | \( 1 - 7.62T + 17T^{2} \) |
| 19 | \( 1 + 3.87T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 6.75T + 29T^{2} \) |
| 31 | \( 1 + 5.01T + 31T^{2} \) |
| 37 | \( 1 - 7.49T + 37T^{2} \) |
| 41 | \( 1 + 7.13T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 9.64T + 47T^{2} \) |
| 53 | \( 1 + 2.03T + 53T^{2} \) |
| 59 | \( 1 - 8.14T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 - 6.37T + 79T^{2} \) |
| 83 | \( 1 - 9.88T + 83T^{2} \) |
| 89 | \( 1 + 7.55T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
| 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
−8.049255877058216777232105414576, −7.51435684851994862435480545352, −6.34402580973059160855131443760, −5.87704839794420840968319458220, −5.11986581690673510815698376781, −4.00320700652485095052377545559, −3.57804178414293863278019978913, −2.69314729699106136576210786758, −1.51844726640622959260073362442, 0,
1.51844726640622959260073362442, 2.69314729699106136576210786758, 3.57804178414293863278019978913, 4.00320700652485095052377545559, 5.11986581690673510815698376781, 5.87704839794420840968319458220, 6.34402580973059160855131443760, 7.51435684851994862435480545352, 8.049255877058216777232105414576
