| L(s) = 1 | − 1.41·2-s + 2.28·5-s + 0.236·7-s + 2.82·8-s − 3.23·10-s − 5.45·11-s − 5.47·13-s − 0.333·14-s − 4.00·16-s + 5.11·17-s − 6.23·19-s + 7.70·22-s − 0.333·23-s + 0.236·25-s + 7.73·26-s + 3.49·29-s − 31-s − 7.23·34-s + 0.540·35-s + 4.70·37-s + 8.81·38-s + 6.47·40-s + 2.62·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 1.02·5-s + 0.0892·7-s + 0.999·8-s − 1.02·10-s − 1.64·11-s − 1.51·13-s − 0.0892·14-s − 1.00·16-s + 1.24·17-s − 1.43·19-s + 1.64·22-s − 0.0696·23-s + 0.0472·25-s + 1.51·26-s + 0.649·29-s − 0.179·31-s − 1.24·34-s + 0.0913·35-s + 0.774·37-s + 1.43·38-s + 1.02·40-s + 0.409·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 5.45T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 + 0.333T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 2.62T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 6.53T + 47T^{2} \) |
| 53 | \( 1 + 1.20T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 6.70T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 0.236T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 4.44T + 83T^{2} \) |
| 89 | \( 1 - 9.35T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
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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
−9.995401955217144931278137741994, −9.094386827249688649575382298352, −7.980722118552733703366298960823, −7.67342463118610725868725653953, −6.38447931459865660232283259981, −5.27889972762175724421981551365, −4.64230219489280958320782283712, −2.78604300376584708259571826407, −1.80380732156626667194330458380, 0,
1.80380732156626667194330458380, 2.78604300376584708259571826407, 4.64230219489280958320782283712, 5.27889972762175724421981551365, 6.38447931459865660232283259981, 7.67342463118610725868725653953, 7.980722118552733703366298960823, 9.094386827249688649575382298352, 9.995401955217144931278137741994
