| L(s) = 1 | + 0.342·2-s − 1.88·4-s + 2.14·5-s − 7-s − 1.32·8-s + 0.735·10-s + 1.42·13-s − 0.342·14-s + 3.31·16-s − 1.38·17-s − 2.59·19-s − 4.04·20-s + 0.0906·23-s − 0.381·25-s + 0.488·26-s + 1.88·28-s + 7.74·29-s − 4.82·31-s + 3.78·32-s − 0.473·34-s − 2.14·35-s − 0.0702·37-s − 0.886·38-s − 2.85·40-s − 6.70·41-s + 6.28·43-s + 0.0309·46-s + ⋯ |
| L(s) = 1 | + 0.241·2-s − 0.941·4-s + 0.961·5-s − 0.377·7-s − 0.469·8-s + 0.232·10-s + 0.396·13-s − 0.0914·14-s + 0.827·16-s − 0.335·17-s − 0.594·19-s − 0.904·20-s + 0.0188·23-s − 0.0763·25-s + 0.0958·26-s + 0.355·28-s + 1.43·29-s − 0.867·31-s + 0.669·32-s − 0.0811·34-s − 0.363·35-s − 0.0115·37-s − 0.143·38-s − 0.451·40-s − 1.04·41-s + 0.958·43-s + 0.00457·46-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.342T + 2T^{2} \) |
| 5 | \( 1 - 2.14T + 5T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 + 1.38T + 17T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 - 0.0906T + 23T^{2} \) |
| 29 | \( 1 - 7.74T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + 0.0702T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 - 6.28T + 43T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 + 3.73T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 - 7.09T + 67T^{2} \) |
| 71 | \( 1 - 8.96T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 - 9.90T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 5.60T + 97T^{2} \) |
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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
−7.57111521012812284836777085155, −6.49034093527710402096602479865, −6.20776382586985658251694784355, −5.35746041421052412955129569262, −4.78255578415672473652195289927, −3.96065798355095831757810800825, −3.22089484310924143044302492448, −2.28356877965318341220820997609, −1.27020513495017404777765606055, 0,
1.27020513495017404777765606055, 2.28356877965318341220820997609, 3.22089484310924143044302492448, 3.96065798355095831757810800825, 4.78255578415672473652195289927, 5.35746041421052412955129569262, 6.20776382586985658251694784355, 6.49034093527710402096602479865, 7.57111521012812284836777085155
