| L(s) = 1 | − 0.517·2-s − 1.73·4-s − 5-s + 1.93·8-s + 0.517·10-s − 3.86·11-s + 1.03·13-s + 2.46·16-s + 1.46·17-s + 2.44·19-s + 1.73·20-s + 1.99·22-s + 1.41·23-s + 25-s − 0.535·26-s + 7.72·29-s − 2.44·31-s − 5.13·32-s − 0.757·34-s + 4.92·37-s − 1.26·38-s − 1.93·40-s − 0.535·41-s − 8.39·43-s + 6.69·44-s − 0.732·46-s − 12.9·47-s + ⋯ |
| L(s) = 1 | − 0.366·2-s − 0.866·4-s − 0.447·5-s + 0.683·8-s + 0.163·10-s − 1.16·11-s + 0.287·13-s + 0.616·16-s + 0.355·17-s + 0.561·19-s + 0.387·20-s + 0.426·22-s + 0.294·23-s + 0.200·25-s − 0.105·26-s + 1.43·29-s − 0.439·31-s − 0.908·32-s − 0.129·34-s + 0.810·37-s − 0.205·38-s − 0.305·40-s − 0.0836·41-s − 1.27·43-s + 1.00·44-s − 0.107·46-s − 1.88·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.517T + 2T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 + 0.535T + 41T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 4.62T + 71T^{2} \) |
| 73 | \( 1 - 6.69T + 73T^{2} \) |
| 79 | \( 1 + 0.535T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.379419037542596438216181730150, −8.186318936218476262283170160130, −7.36622076828102584204474547365, −6.36448755947280954930954949971, −5.20519348519371306120417083344, −4.81039108551313257287211212701, −3.69833325696859371517139020562, −2.85404930910861554153089382108, −1.28866959088606292544524932715, 0,
1.28866959088606292544524932715, 2.85404930910861554153089382108, 3.69833325696859371517139020562, 4.81039108551313257287211212701, 5.20519348519371306120417083344, 6.36448755947280954930954949971, 7.36622076828102584204474547365, 8.186318936218476262283170160130, 8.379419037542596438216181730150
