| L(s) = 1 | − 2-s − 3.01·3-s + 4-s + 0.160·5-s + 3.01·6-s − 7-s − 8-s + 6.09·9-s − 0.160·10-s − 2.88·11-s − 3.01·12-s + 6.63·13-s + 14-s − 0.484·15-s + 16-s + 2.24·17-s − 6.09·18-s + 1.67·19-s + 0.160·20-s + 3.01·21-s + 2.88·22-s − 2.65·23-s + 3.01·24-s − 4.97·25-s − 6.63·26-s − 9.31·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.74·3-s + 0.5·4-s + 0.0719·5-s + 1.23·6-s − 0.377·7-s − 0.353·8-s + 2.03·9-s − 0.0508·10-s − 0.869·11-s − 0.870·12-s + 1.84·13-s + 0.267·14-s − 0.125·15-s + 0.250·16-s + 0.543·17-s − 1.43·18-s + 0.383·19-s + 0.0359·20-s + 0.657·21-s + 0.614·22-s − 0.553·23-s + 0.615·24-s − 0.994·25-s − 1.30·26-s − 1.79·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 3.01T + 3T^{2} \) |
| 5 | \( 1 - 0.160T + 5T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 - 5.45T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 - 5.83T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 - 3.19T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 7.51T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−7.78235180467133999974746793480, −6.79776633838169732687489604593, −6.30816791613583845335767267261, −5.68127774218487566384472520255, −5.22712299314705494282272905050, −4.07494102554088734334225817296, −3.30731093030002735064805526793, −1.88379951528299611558463885400, −0.991358447280782070904792707750, 0,
0.991358447280782070904792707750, 1.88379951528299611558463885400, 3.30731093030002735064805526793, 4.07494102554088734334225817296, 5.22712299314705494282272905050, 5.68127774218487566384472520255, 6.30816791613583845335767267261, 6.79776633838169732687489604593, 7.78235180467133999974746793480
