L(s) = 1 | + 2-s + 4-s + 0.102·5-s + 0.350·7-s + 8-s + 0.102·10-s + 2.02·11-s − 0.0298·13-s + 0.350·14-s + 16-s − 3.57·17-s − 2.36·19-s + 0.102·20-s + 2.02·22-s − 3.59·23-s − 4.98·25-s − 0.0298·26-s + 0.350·28-s − 4.55·29-s + 0.839·31-s + 32-s − 3.57·34-s + 0.0357·35-s − 3.75·37-s − 2.36·38-s + 0.102·40-s + 1.43·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.0456·5-s + 0.132·7-s + 0.353·8-s + 0.0322·10-s + 0.611·11-s − 0.00828·13-s + 0.0936·14-s + 0.250·16-s − 0.867·17-s − 0.542·19-s + 0.0228·20-s + 0.432·22-s − 0.748·23-s − 0.997·25-s − 0.00586·26-s + 0.0662·28-s − 0.846·29-s + 0.150·31-s + 0.176·32-s − 0.613·34-s + 0.00604·35-s − 0.617·37-s − 0.383·38-s + 0.0161·40-s + 0.224·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 0.102T + 5T^{2} \) |
| 7 | \( 1 - 0.350T + 7T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 13 | \( 1 + 0.0298T + 13T^{2} \) |
| 17 | \( 1 + 3.57T + 17T^{2} \) |
| 19 | \( 1 + 2.36T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 + 4.55T + 29T^{2} \) |
| 31 | \( 1 - 0.839T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 + 9.52T + 53T^{2} \) |
| 59 | \( 1 + 9.78T + 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 - 9.49T + 71T^{2} \) |
| 73 | \( 1 - 7.96T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 + 9.62T + 83T^{2} \) |
| 89 | \( 1 + 0.871T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.45719412184687954717431718426, −6.50747134593484904142325304853, −6.22552702785383277954884296269, −5.35451749543796650488997770517, −4.57091469481466709530258544850, −3.98593296197757115006702589612, −3.26782205360561070293066218534, −2.19290692189994986296235755555, −1.59372808277006851229008052603, 0,
1.59372808277006851229008052603, 2.19290692189994986296235755555, 3.26782205360561070293066218534, 3.98593296197757115006702589612, 4.57091469481466709530258544850, 5.35451749543796650488997770517, 6.22552702785383277954884296269, 6.50747134593484904142325304853, 7.45719412184687954717431718426