| L(s) = 1 | + 3-s + 3.23·5-s + 3.23·7-s + 9-s + 2·11-s − 13-s + 3.23·15-s − 4.47·17-s + 0.763·19-s + 3.23·21-s + 6.47·23-s + 5.47·25-s + 27-s − 4.47·29-s − 5.70·31-s + 2·33-s + 10.4·35-s − 8.47·37-s − 39-s − 3.23·41-s − 2.47·43-s + 3.23·45-s − 10.9·47-s + 3.47·49-s − 4.47·51-s − 0.472·53-s + 6.47·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.44·5-s + 1.22·7-s + 0.333·9-s + 0.603·11-s − 0.277·13-s + 0.835·15-s − 1.08·17-s + 0.175·19-s + 0.706·21-s + 1.34·23-s + 1.09·25-s + 0.192·27-s − 0.830·29-s − 1.02·31-s + 0.348·33-s + 1.77·35-s − 1.39·37-s − 0.160·39-s − 0.505·41-s − 0.376·43-s + 0.482·45-s − 1.59·47-s + 0.496·49-s − 0.626·51-s − 0.0648·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.971282035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.971282035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 3.23T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 0.472T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−9.419270198857089720855464717727, −9.073996327171814837263654195291, −8.237289777356480465113995409215, −7.18105716137131526926880243147, −6.47573982350194907663572989312, −5.30855473708207105134046313390, −4.77244537814582798848949496205, −3.44104403666905601806020929103, −2.12071538571841718057557201101, −1.57261524429216743599767558902,
1.57261524429216743599767558902, 2.12071538571841718057557201101, 3.44104403666905601806020929103, 4.77244537814582798848949496205, 5.30855473708207105134046313390, 6.47573982350194907663572989312, 7.18105716137131526926880243147, 8.237289777356480465113995409215, 9.073996327171814837263654195291, 9.419270198857089720855464717727
