| L(s) = 1 | − 4.22·2-s − 3·3-s + 9.85·4-s − 5.85·5-s + 12.6·6-s − 24.1·7-s − 7.85·8-s + 9·9-s + 24.7·10-s − 33.8·11-s − 29.5·12-s + 101.·14-s + 17.5·15-s − 45.6·16-s − 49.3·17-s − 38.0·18-s − 76.8·19-s − 57.7·20-s + 72.3·21-s + 143.·22-s + 6.29·23-s + 23.5·24-s − 90.6·25-s − 27·27-s − 237.·28-s + 100.·29-s − 74.2·30-s + ⋯ |
| L(s) = 1 | − 1.49·2-s − 0.577·3-s + 1.23·4-s − 0.524·5-s + 0.862·6-s − 1.30·7-s − 0.347·8-s + 0.333·9-s + 0.783·10-s − 0.928·11-s − 0.711·12-s + 1.94·14-s + 0.302·15-s − 0.713·16-s − 0.704·17-s − 0.498·18-s − 0.927·19-s − 0.645·20-s + 0.752·21-s + 1.38·22-s + 0.0570·23-s + 0.200·24-s − 0.725·25-s − 0.192·27-s − 1.60·28-s + 0.646·29-s − 0.452·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02029435312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02029435312\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.22T + 8T^{2} \) |
| 5 | \( 1 + 5.85T + 125T^{2} \) |
| 7 | \( 1 + 24.1T + 343T^{2} \) |
| 11 | \( 1 + 33.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 49.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 6.29T + 1.21e4T^{2} \) |
| 29 | \( 1 - 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 307.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 76.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 514.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 67.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 25.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 588.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 895.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 968.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 119.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 480.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 16.6T + 9.12e5T^{2} \) |
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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
−10.38080556267973448672646436385, −9.681977452921700544754833856025, −8.776785862408463347669824515035, −7.928407378000778678703276943265, −6.97584531486291022348372984145, −6.31018019640533137155931364863, −4.88945296791598976211258030733, −3.46561745234454834440928875626, −1.96646446141572847860779562592, −0.10686097261411373804228766783,
0.10686097261411373804228766783, 1.96646446141572847860779562592, 3.46561745234454834440928875626, 4.88945296791598976211258030733, 6.31018019640533137155931364863, 6.97584531486291022348372984145, 7.928407378000778678703276943265, 8.776785862408463347669824515035, 9.681977452921700544754833856025, 10.38080556267973448672646436385
