L(s) = 1 | + 2.86·3-s − 1.20·5-s + 5.22·9-s − 0.847·11-s + 1.77·13-s − 3.44·15-s − 1.33·17-s − 5.12·19-s − 0.816·23-s − 3.55·25-s + 6.36·27-s − 8.14·29-s − 5.88·31-s − 2.42·33-s + 1.65·37-s + 5.09·39-s − 41-s − 3.82·43-s − 6.27·45-s − 4.67·47-s − 3.81·51-s − 3.64·53-s + 1.01·55-s − 14.6·57-s + 0.851·59-s + 9.13·61-s − 2.13·65-s + ⋯ |
L(s) = 1 | + 1.65·3-s − 0.537·5-s + 1.74·9-s − 0.255·11-s + 0.492·13-s − 0.889·15-s − 0.322·17-s − 1.17·19-s − 0.170·23-s − 0.711·25-s + 1.22·27-s − 1.51·29-s − 1.05·31-s − 0.422·33-s + 0.272·37-s + 0.815·39-s − 0.156·41-s − 0.583·43-s − 0.935·45-s − 0.681·47-s − 0.534·51-s − 0.501·53-s + 0.137·55-s − 1.94·57-s + 0.110·59-s + 1.16·61-s − 0.264·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 11 | \( 1 + 0.847T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 0.816T + 23T^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 31 | \( 1 + 5.88T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 43 | \( 1 + 3.82T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 - 0.851T + 59T^{2} \) |
| 61 | \( 1 - 9.13T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 + 3.08T + 73T^{2} \) |
| 79 | \( 1 - 8.12T + 79T^{2} \) |
| 83 | \( 1 + 3.90T + 83T^{2} \) |
| 89 | \( 1 - 9.52T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−7.70153682093312907382654791073, −7.06049195277225823620075470955, −6.25163326586163728172120237161, −5.31470840278884621438739172841, −4.25516582417236055124219091736, −3.82717393796372414264892618471, −3.18594246070939521299373051815, −2.22812907272853874843382396641, −1.67409145253713901857800686911, 0,
1.67409145253713901857800686911, 2.22812907272853874843382396641, 3.18594246070939521299373051815, 3.82717393796372414264892618471, 4.25516582417236055124219091736, 5.31470840278884621438739172841, 6.25163326586163728172120237161, 7.06049195277225823620075470955, 7.70153682093312907382654791073