L(s) = 1 | + 3-s − 2.57·5-s − 1.35·7-s + 9-s + 4.35·11-s + 4.79·13-s − 2.57·15-s + 6.79·17-s + 1.64·19-s − 1.35·21-s + 3.64·23-s + 1.64·25-s + 27-s − 4.35·29-s + 31-s + 4.35·33-s + 3.50·35-s + 1.42·37-s + 4.79·39-s + 2.71·41-s + 5.07·43-s − 2.57·45-s − 1.35·47-s − 5.15·49-s + 6.79·51-s − 8.21·53-s − 11.2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.15·5-s − 0.513·7-s + 0.333·9-s + 1.31·11-s + 1.33·13-s − 0.665·15-s + 1.64·17-s + 0.376·19-s − 0.296·21-s + 0.759·23-s + 0.328·25-s + 0.192·27-s − 0.809·29-s + 0.179·31-s + 0.758·33-s + 0.591·35-s + 0.233·37-s + 0.767·39-s + 0.424·41-s + 0.774·43-s − 0.384·45-s − 0.198·47-s − 0.736·49-s + 0.951·51-s − 1.12·53-s − 1.51·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.476836969\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.476836969\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 1.42T + 37T^{2} \) |
| 41 | \( 1 - 2.71T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 + 7.01T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 2.07T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 - 0.717T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 0.361T + 97T^{2} \) |
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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
−7.75596077441015913031566462529, −7.39605804520452688282819923287, −6.43275343254887133840160880612, −5.96722697178826307948596422712, −4.86599968169451553761425036477, −3.97345816742888616208364746593, −3.50190763762931604180974984880, −3.09492403690426209251539319901, −1.58318777938486975447371674613, −0.825265481505541245147233901775,
0.825265481505541245147233901775, 1.58318777938486975447371674613, 3.09492403690426209251539319901, 3.50190763762931604180974984880, 3.97345816742888616208364746593, 4.86599968169451553761425036477, 5.96722697178826307948596422712, 6.43275343254887133840160880612, 7.39605804520452688282819923287, 7.75596077441015913031566462529