| L(s) = 1 | + 5.38·2-s + 8.61·3-s − 2.96·4-s + 25·5-s + 46.4·6-s − 158.·7-s − 188.·8-s − 168.·9-s + 134.·10-s + 121·11-s − 25.4·12-s − 522.·13-s − 853.·14-s + 215.·15-s − 920.·16-s − 1.46e3·17-s − 909.·18-s + 361·19-s − 74.0·20-s − 1.36e3·21-s + 652.·22-s + 2.55e3·23-s − 1.62e3·24-s + 625·25-s − 2.81e3·26-s − 3.54e3·27-s + 468.·28-s + ⋯ |
| L(s) = 1 | + 0.952·2-s + 0.552·3-s − 0.0925·4-s + 0.447·5-s + 0.526·6-s − 1.22·7-s − 1.04·8-s − 0.694·9-s + 0.426·10-s + 0.301·11-s − 0.0511·12-s − 0.858·13-s − 1.16·14-s + 0.247·15-s − 0.898·16-s − 1.23·17-s − 0.661·18-s + 0.229·19-s − 0.0413·20-s − 0.674·21-s + 0.287·22-s + 1.00·23-s − 0.574·24-s + 0.200·25-s − 0.817·26-s − 0.936·27-s + 0.113·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.157100299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.157100299\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 5.38T + 32T^{2} \) |
| 3 | \( 1 - 8.61T + 243T^{2} \) |
| 7 | \( 1 + 158.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 522.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.46e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 250.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 731.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.55e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.14e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.73e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.33e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.29e4T + 8.58e9T^{2} \) |
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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
−9.198801783715524440346253713718, −8.667889142869755993637870364306, −7.30157635800994072745458237195, −6.43490526328782989528123965216, −5.74807374900684664109668862419, −4.81028676781895498649852857194, −3.84454628274575872079562706641, −2.93544672333016497330511806301, −2.38938658679717735124445363532, −0.49993249429340825175686161839,
0.49993249429340825175686161839, 2.38938658679717735124445363532, 2.93544672333016497330511806301, 3.84454628274575872079562706641, 4.81028676781895498649852857194, 5.74807374900684664109668862419, 6.43490526328782989528123965216, 7.30157635800994072745458237195, 8.667889142869755993637870364306, 9.198801783715524440346253713718
