L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 2·7-s − 2·9-s − 2·10-s − 11-s − 2·12-s + 4·14-s + 15-s − 4·16-s − 2·17-s − 4·18-s − 2·20-s − 2·21-s − 2·22-s − 23-s − 4·25-s + 5·27-s + 4·28-s + 2·30-s − 7·31-s − 8·32-s + 33-s − 4·34-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s − 2/3·9-s − 0.632·10-s − 0.301·11-s − 0.577·12-s + 1.06·14-s + 0.258·15-s − 16-s − 0.485·17-s − 0.942·18-s − 0.447·20-s − 0.436·21-s − 0.426·22-s − 0.208·23-s − 4/5·25-s + 0.962·27-s + 0.755·28-s + 0.365·30-s − 1.25·31-s − 1.41·32-s + 0.174·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.679083346749551349707825193313, −7.972094664118160382294418175959, −6.98822017919486224475963260290, −6.13551737647869362536665067533, −5.42467290200223196482028296291, −4.84072493234829119227722123670, −4.01156608984212096591011250141, −3.11506443817379691538930485019, −1.98576135654663457689533375342, 0,
1.98576135654663457689533375342, 3.11506443817379691538930485019, 4.01156608984212096591011250141, 4.84072493234829119227722123670, 5.42467290200223196482028296291, 6.13551737647869362536665067533, 6.98822017919486224475963260290, 7.972094664118160382294418175959, 8.679083346749551349707825193313