| L(s) = 1 | − 0.823·2-s − 1.32·4-s − 2.98·5-s − 7-s + 2.73·8-s + 2.45·10-s − 2.20·13-s + 0.823·14-s + 0.390·16-s + 4.40·17-s + 1.72·19-s + 3.94·20-s + 8.39·23-s + 3.91·25-s + 1.81·26-s + 1.32·28-s + 3.29·29-s + 7.47·31-s − 5.79·32-s − 3.62·34-s + 2.98·35-s − 8.78·37-s − 1.42·38-s − 8.16·40-s + 5.39·41-s − 9.44·43-s − 6.91·46-s + ⋯ |
| L(s) = 1 | − 0.582·2-s − 0.660·4-s − 1.33·5-s − 0.377·7-s + 0.967·8-s + 0.777·10-s − 0.610·13-s + 0.220·14-s + 0.0976·16-s + 1.06·17-s + 0.395·19-s + 0.882·20-s + 1.75·23-s + 0.782·25-s + 0.355·26-s + 0.249·28-s + 0.611·29-s + 1.34·31-s − 1.02·32-s − 0.622·34-s + 0.504·35-s − 1.44·37-s − 0.230·38-s − 1.29·40-s + 0.842·41-s − 1.44·43-s − 1.01·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6620471775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6620471775\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.823T + 2T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 13 | \( 1 + 2.20T + 13T^{2} \) |
| 17 | \( 1 - 4.40T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 8.39T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 - 5.39T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 - 5.39T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 + 3.47T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 4.40T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 7.18T + 79T^{2} \) |
| 83 | \( 1 - 7.63T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−7.990315285511557673722488539454, −7.31854063097310586842802246058, −6.85958674527943316310106163032, −5.69375926781869913615308078146, −4.81972173216196350685192980785, −4.46261210479154778376408191781, −3.39229036964190605357192011005, −3.00832110437892424043301089536, −1.36823596779537901045133446312, −0.49524732458707039051476159072,
0.49524732458707039051476159072, 1.36823596779537901045133446312, 3.00832110437892424043301089536, 3.39229036964190605357192011005, 4.46261210479154778376408191781, 4.81972173216196350685192980785, 5.69375926781869913615308078146, 6.85958674527943316310106163032, 7.31854063097310586842802246058, 7.990315285511557673722488539454
