| L(s) = 1 | − 2-s + 3-s + 4-s − 0.888·5-s − 6-s + 1.76·7-s − 8-s + 9-s + 0.888·10-s + 6.55·11-s + 12-s − 3.30·13-s − 1.76·14-s − 0.888·15-s + 16-s + 3.40·17-s − 18-s − 5.20·19-s − 0.888·20-s + 1.76·21-s − 6.55·22-s − 7.41·23-s − 24-s − 4.20·25-s + 3.30·26-s + 27-s + 1.76·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.397·5-s − 0.408·6-s + 0.666·7-s − 0.353·8-s + 0.333·9-s + 0.281·10-s + 1.97·11-s + 0.288·12-s − 0.915·13-s − 0.471·14-s − 0.229·15-s + 0.250·16-s + 0.826·17-s − 0.235·18-s − 1.19·19-s − 0.198·20-s + 0.384·21-s − 1.39·22-s − 1.54·23-s − 0.204·24-s − 0.841·25-s + 0.647·26-s + 0.192·27-s + 0.333·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 + 0.888T + 5T^{2} \) |
| 7 | \( 1 - 1.76T + 7T^{2} \) |
| 11 | \( 1 - 6.55T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 37 | \( 1 + 6.52T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 + 8.79T + 47T^{2} \) |
| 53 | \( 1 + 5.06T + 53T^{2} \) |
| 59 | \( 1 + 9.14T + 59T^{2} \) |
| 61 | \( 1 - 0.160T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 2.74T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 0.825T + 89T^{2} \) |
| 97 | \( 1 - 0.815T + 97T^{2} \) |
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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
−7.985916417967283223911624221369, −7.19581341053458149186141683327, −6.57317183073842193484368383539, −5.79691578790525626542907422032, −4.64242427019056965434433017121, −3.96851942248384403393433091246, −3.28437599092829611530885521049, −1.93951965578979330375919762005, −1.57819149758067064626497707770, 0,
1.57819149758067064626497707770, 1.93951965578979330375919762005, 3.28437599092829611530885521049, 3.96851942248384403393433091246, 4.64242427019056965434433017121, 5.79691578790525626542907422032, 6.57317183073842193484368383539, 7.19581341053458149186141683327, 7.985916417967283223911624221369
