| L(s) = 1 | + 8.40·2-s − 9·3-s + 38.5·4-s + 77.4·5-s − 75.6·6-s + 52.6·7-s + 55.2·8-s + 81·9-s + 650.·10-s + 723.·11-s − 347.·12-s − 866.·13-s + 442.·14-s − 697.·15-s − 770.·16-s + 1.18e3·17-s + 680.·18-s + 2.70e3·19-s + 2.98e3·20-s − 474.·21-s + 6.08e3·22-s − 4.71e3·23-s − 497.·24-s + 2.87e3·25-s − 7.27e3·26-s − 729·27-s + 2.03e3·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 0.577·3-s + 1.20·4-s + 1.38·5-s − 0.857·6-s + 0.406·7-s + 0.305·8-s + 0.333·9-s + 2.05·10-s + 1.80·11-s − 0.696·12-s − 1.42·13-s + 0.603·14-s − 0.799·15-s − 0.752·16-s + 0.991·17-s + 0.495·18-s + 1.71·19-s + 1.67·20-s − 0.234·21-s + 2.67·22-s − 1.86·23-s − 0.176·24-s + 0.919·25-s − 2.11·26-s − 0.192·27-s + 0.490·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.744785148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.744785148\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 103 | \( 1 - 1.06e4T \) |
| good | 2 | \( 1 - 8.40T + 32T^{2} \) |
| 5 | \( 1 - 77.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 52.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 723.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 866.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.29e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.90e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.07e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.30e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.48e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.57e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.87e4T + 8.58e9T^{2} \) |
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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
−11.28783319345048203011772354003, −9.771585782322198314053242608202, −9.493750453923955998481579073659, −7.53685597408008109536693497644, −6.38959576262985245713349393537, −5.76063149805846099248403104566, −4.96605045654613108659082252648, −3.90003636489777826516019260009, −2.46036754365588865599730330702, −1.24898366436183763078235564005,
1.24898366436183763078235564005, 2.46036754365588865599730330702, 3.90003636489777826516019260009, 4.96605045654613108659082252648, 5.76063149805846099248403104566, 6.38959576262985245713349393537, 7.53685597408008109536693497644, 9.493750453923955998481579073659, 9.771585782322198314053242608202, 11.28783319345048203011772354003
