| L(s) = 1 | − 2-s + 2.37·3-s + 4-s − 2.37·6-s − 7-s − 8-s + 2.65·9-s + 1.02·11-s + 2.37·12-s + 1.72·13-s + 14-s + 16-s − 17-s − 2.65·18-s − 7.22·19-s − 2.37·21-s − 1.02·22-s − 5.13·23-s − 2.37·24-s − 1.72·26-s − 0.829·27-s − 28-s + 5.40·29-s − 7.47·31-s − 32-s + 2.44·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.37·3-s + 0.5·4-s − 0.970·6-s − 0.377·7-s − 0.353·8-s + 0.883·9-s + 0.310·11-s + 0.686·12-s + 0.478·13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s − 0.624·18-s − 1.65·19-s − 0.518·21-s − 0.219·22-s − 1.07·23-s − 0.485·24-s − 0.338·26-s − 0.159·27-s − 0.188·28-s + 1.00·29-s − 1.34·31-s − 0.176·32-s + 0.425·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.37T + 3T^{2} \) |
| 11 | \( 1 - 1.02T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 19 | \( 1 + 7.22T + 19T^{2} \) |
| 23 | \( 1 + 5.13T + 23T^{2} \) |
| 29 | \( 1 - 5.40T + 29T^{2} \) |
| 31 | \( 1 + 7.47T + 31T^{2} \) |
| 37 | \( 1 - 8.84T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 + 0.932T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 0.679T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 7.19T + 73T^{2} \) |
| 79 | \( 1 + 2.27T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.096484188642323822944193526913, −7.22838326373966906439697263110, −6.44025138610987950537552852693, −5.93754649242718136109872602163, −4.54798521107767314863271886181, −3.83261430818539476534190782906, −3.08514806289706101380564092136, −2.26172032649799732627107586453, −1.56942841437104968716130794372, 0,
1.56942841437104968716130794372, 2.26172032649799732627107586453, 3.08514806289706101380564092136, 3.83261430818539476534190782906, 4.54798521107767314863271886181, 5.93754649242718136109872602163, 6.44025138610987950537552852693, 7.22838326373966906439697263110, 8.096484188642323822944193526913
