| L(s) = 1 | − 1.49·2-s + 3-s + 0.245·4-s − 1.49·6-s − 3.73·7-s + 2.62·8-s + 9-s + 4.52·11-s + 0.245·12-s − 6.66·13-s + 5.59·14-s − 4.43·16-s − 1.58·17-s − 1.49·18-s − 0.710·19-s − 3.73·21-s − 6.78·22-s + 8.96·23-s + 2.62·24-s + 9.98·26-s + 27-s − 0.917·28-s + 1.28·29-s + 6.01·31-s + 1.38·32-s + 4.52·33-s + 2.38·34-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 0.577·3-s + 0.122·4-s − 0.611·6-s − 1.41·7-s + 0.929·8-s + 0.333·9-s + 1.36·11-s + 0.0709·12-s − 1.84·13-s + 1.49·14-s − 1.10·16-s − 0.385·17-s − 0.353·18-s − 0.162·19-s − 0.814·21-s − 1.44·22-s + 1.87·23-s + 0.536·24-s + 1.95·26-s + 0.192·27-s − 0.173·28-s + 0.238·29-s + 1.08·31-s + 0.244·32-s + 0.787·33-s + 0.408·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.49T + 2T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 + 6.66T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 + 0.710T + 19T^{2} \) |
| 23 | \( 1 - 8.96T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 + 4.78T + 37T^{2} \) |
| 41 | \( 1 + 7.22T + 41T^{2} \) |
| 43 | \( 1 - 4.67T + 43T^{2} \) |
| 53 | \( 1 - 2.17T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 9.91T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + 7.73T + 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−8.523543697447016589028545493263, −7.36735062100642213348885796285, −7.02336116360206292584754344938, −6.37552863015305631044114735202, −4.99533861054987839656388590600, −4.30140987598686653368462002456, −3.27103814731721413452569288915, −2.47174560961267989396799535597, −1.22706003032103126444584575843, 0,
1.22706003032103126444584575843, 2.47174560961267989396799535597, 3.27103814731721413452569288915, 4.30140987598686653368462002456, 4.99533861054987839656388590600, 6.37552863015305631044114735202, 7.02336116360206292584754344938, 7.36735062100642213348885796285, 8.523543697447016589028545493263
