| L(s) = 1 | + 1.27·2-s + 3-s − 0.363·4-s + 5-s + 1.27·6-s − 2.12·7-s − 3.02·8-s + 9-s + 1.27·10-s + 4.20·11-s − 0.363·12-s − 0.372·13-s − 2.72·14-s + 15-s − 3.14·16-s − 2.93·17-s + 1.27·18-s − 3.42·19-s − 0.363·20-s − 2.12·21-s + 5.37·22-s − 3.02·24-s + 25-s − 0.476·26-s + 27-s + 0.773·28-s + 5.53·29-s + ⋯ |
| L(s) = 1 | + 0.904·2-s + 0.577·3-s − 0.181·4-s + 0.447·5-s + 0.522·6-s − 0.803·7-s − 1.06·8-s + 0.333·9-s + 0.404·10-s + 1.26·11-s − 0.104·12-s − 0.103·13-s − 0.727·14-s + 0.258·15-s − 0.785·16-s − 0.711·17-s + 0.301·18-s − 0.786·19-s − 0.0812·20-s − 0.464·21-s + 1.14·22-s − 0.617·24-s + 0.200·25-s − 0.0933·26-s + 0.192·27-s + 0.146·28-s + 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 29 | \( 1 - 5.53T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 + 3.81T + 37T^{2} \) |
| 41 | \( 1 + 6.57T + 41T^{2} \) |
| 43 | \( 1 - 0.320T + 43T^{2} \) |
| 47 | \( 1 - 5.48T + 47T^{2} \) |
| 53 | \( 1 + 4.21T + 53T^{2} \) |
| 59 | \( 1 - 9.18T + 59T^{2} \) |
| 61 | \( 1 + 5.44T + 61T^{2} \) |
| 67 | \( 1 - 5.77T + 67T^{2} \) |
| 71 | \( 1 + 8.88T + 71T^{2} \) |
| 73 | \( 1 + 7.35T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 3.05T + 83T^{2} \) |
| 89 | \( 1 + 3.04T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−7.20019263243866508315186175654, −6.62541277568993450543008982658, −6.16021808874954192617614857255, −5.34817094516511290044597868968, −4.47212989789842405091484225311, −3.94917967943983616865193638177, −3.26133694085002494496040417969, −2.51385711153643224638146413928, −1.50167917672363093085628128010, 0,
1.50167917672363093085628128010, 2.51385711153643224638146413928, 3.26133694085002494496040417969, 3.94917967943983616865193638177, 4.47212989789842405091484225311, 5.34817094516511290044597868968, 6.16021808874954192617614857255, 6.62541277568993450543008982658, 7.20019263243866508315186175654
