| L(s) = 1 | − 1.70i·2-s + 1.38i·3-s − 0.922·4-s + (−1.53 − 1.62i)5-s + 2.36·6-s − 5.05i·7-s − 1.84i·8-s + 1.08·9-s + (−2.77 + 2.62i)10-s − 5.25·11-s − 1.27i·12-s − 2.84i·13-s − 8.64·14-s + (2.24 − 2.12i)15-s − 4.99·16-s + 6.16i·17-s + ⋯ |
| L(s) = 1 | − 1.20i·2-s + 0.798i·3-s − 0.461·4-s + (−0.686 − 0.726i)5-s + 0.965·6-s − 1.91i·7-s − 0.651i·8-s + 0.362·9-s + (−0.878 + 0.830i)10-s − 1.58·11-s − 0.368i·12-s − 0.788i·13-s − 2.31·14-s + (0.580 − 0.548i)15-s − 1.24·16-s + 1.49i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7919508650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7919508650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.53 + 1.62i)T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70iT - 2T^{2} \) |
| 3 | \( 1 - 1.38iT - 3T^{2} \) |
| 7 | \( 1 + 5.05iT - 7T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 13 | \( 1 + 2.84iT - 13T^{2} \) |
| 17 | \( 1 - 6.16iT - 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 - 2.54iT - 23T^{2} \) |
| 29 | \( 1 - 9.53T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 + 8.85iT - 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 - 0.0179iT - 43T^{2} \) |
| 47 | \( 1 - 5.95iT - 47T^{2} \) |
| 53 | \( 1 + 8.07iT - 53T^{2} \) |
| 59 | \( 1 - 3.28T + 59T^{2} \) |
| 61 | \( 1 - 7.19T + 61T^{2} \) |
| 67 | \( 1 - 2.92iT - 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 - 4.29iT - 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 9.27iT - 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−9.768234285174473932470624744711, −8.486784352825010973961487131242, −7.69589465177973493348325704481, −7.01812293553008529986673725084, −5.41993416647185686226873681316, −4.36277337440551730987384234247, −3.97495890137454420685038983486, −3.15837325828961373582472287958, −1.50427081774321952474121072425, −0.33043365729257096277737037503,
2.33952975986904406942527178933, 2.72471933644512459366135659081, 4.71050452073429151584338955398, 5.39342238086627575669856696607, 6.41695002784763167107698245202, 6.87810899723422688969387652779, 7.71521654726518783628638106246, 8.274762838576365574017357562247, 9.010834834677890562504892504647, 10.16667743157031927970583310892
