| L(s) = 1 | + 1.16·2-s − 0.645·4-s + 1.64·5-s − 1.16·7-s − 3.07·8-s + 1.91·10-s − 4.24·13-s − 1.35·14-s − 2.29·16-s + 3.07·17-s + 6.57·19-s − 1.06·20-s − 1.35·23-s − 2.29·25-s − 4.93·26-s + 0.751·28-s − 1.16·29-s − 3.64·31-s + 3.49·32-s + 3.58·34-s − 1.91·35-s − 3.35·37-s + 7.64·38-s − 5.06·40-s + 9.64·41-s − 10.0·43-s − 1.57·46-s + ⋯ |
| L(s) = 1 | + 0.822·2-s − 0.322·4-s + 0.736·5-s − 0.439·7-s − 1.08·8-s + 0.605·10-s − 1.17·13-s − 0.361·14-s − 0.572·16-s + 0.746·17-s + 1.50·19-s − 0.237·20-s − 0.282·23-s − 0.458·25-s − 0.968·26-s + 0.142·28-s − 0.216·29-s − 0.654·31-s + 0.617·32-s + 0.614·34-s − 0.323·35-s − 0.551·37-s + 1.24·38-s − 0.801·40-s + 1.50·41-s − 1.53·43-s − 0.232·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 - 6.57T + 19T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 + 1.16T + 29T^{2} \) |
| 31 | \( 1 + 3.64T + 31T^{2} \) |
| 37 | \( 1 + 3.35T + 37T^{2} \) |
| 41 | \( 1 - 9.64T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 + 1.91T + 61T^{2} \) |
| 67 | \( 1 + 3.64T + 67T^{2} \) |
| 71 | \( 1 - 9.29T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 0.751T + 79T^{2} \) |
| 83 | \( 1 + 8.89T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 5.93T + 97T^{2} \) |
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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
−8.203059929109326551821367268112, −7.45989246605309448295782033318, −6.56573760490337552162626535072, −5.74030798855529643783512247056, −5.27109010938642201559672806147, −4.53283899429102684937723423081, −3.42396333148932154915860757828, −2.90231916332653533981941786503, −1.64032689030855636120003286827, 0,
1.64032689030855636120003286827, 2.90231916332653533981941786503, 3.42396333148932154915860757828, 4.53283899429102684937723423081, 5.27109010938642201559672806147, 5.74030798855529643783512247056, 6.56573760490337552162626535072, 7.45989246605309448295782033318, 8.203059929109326551821367268112
