| L(s) = 1 | + 1.83·2-s − 3-s + 1.35·4-s − 5-s − 1.83·6-s − 7-s − 1.18·8-s + 9-s − 1.83·10-s + 3.84·11-s − 1.35·12-s − 1.18·13-s − 1.83·14-s + 15-s − 4.87·16-s − 2·17-s + 1.83·18-s + 5.22·19-s − 1.35·20-s + 21-s + 7.04·22-s − 23-s + 1.18·24-s + 25-s − 2.17·26-s − 27-s − 1.35·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.577·3-s + 0.677·4-s − 0.447·5-s − 0.747·6-s − 0.377·7-s − 0.417·8-s + 0.333·9-s − 0.579·10-s + 1.15·11-s − 0.391·12-s − 0.329·13-s − 0.489·14-s + 0.258·15-s − 1.21·16-s − 0.485·17-s + 0.431·18-s + 1.19·19-s − 0.303·20-s + 0.218·21-s + 1.50·22-s − 0.208·23-s + 0.240·24-s + 0.200·25-s − 0.426·26-s − 0.192·27-s − 0.256·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 5.22T + 19T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 + 8.42T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 - 1.13T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 0.0668T + 61T^{2} \) |
| 67 | \( 1 + 5.31T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 3.15T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 7.09T + 97T^{2} \) |
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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
−8.670926203800526115085752237141, −7.41635269927490864994137525219, −6.77871846361134919646162177304, −6.12954389496011493067366224824, −5.25732733800074864029017734351, −4.65305621131570211265364789656, −3.70556146365714632774223741622, −3.22787676744749817767575242151, −1.71171149496521530390417611758, 0,
1.71171149496521530390417611758, 3.22787676744749817767575242151, 3.70556146365714632774223741622, 4.65305621131570211265364789656, 5.25732733800074864029017734351, 6.12954389496011493067366224824, 6.77871846361134919646162177304, 7.41635269927490864994137525219, 8.670926203800526115085752237141
