| L(s) = 1 | + (−2.15 + 0.594i)5-s + 0.633i·7-s + 0.177·11-s + i·13-s + 4.48i·17-s + 6.48i·23-s + (4.29 − 2.56i)25-s − 3.49·29-s + (−0.376 − 1.36i)35-s − 1.75i·37-s − 7.67·41-s − 7.49i·43-s − 3.18i·47-s + 6.59·49-s + 1.75i·53-s + ⋯ |
| L(s) = 1 | + (−0.964 + 0.265i)5-s + 0.239i·7-s + 0.0536·11-s + 0.277i·13-s + 1.08i·17-s + 1.35i·23-s + (0.858 − 0.512i)25-s − 0.649·29-s + (−0.0636 − 0.230i)35-s − 0.288i·37-s − 1.19·41-s − 1.14i·43-s − 0.465i·47-s + 0.942·49-s + 0.241i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2571852403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2571852403\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.15 - 0.594i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 0.633iT - 7T^{2} \) |
| 11 | \( 1 - 0.177T + 11T^{2} \) |
| 17 | \( 1 - 4.48iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6.48iT - 23T^{2} \) |
| 29 | \( 1 + 3.49T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.75iT - 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 + 7.49iT - 43T^{2} \) |
| 47 | \( 1 + 3.18iT - 47T^{2} \) |
| 53 | \( 1 - 1.75iT - 53T^{2} \) |
| 59 | \( 1 - 3.93T + 59T^{2} \) |
| 61 | \( 1 - 3.01T + 61T^{2} \) |
| 67 | \( 1 - 1.62iT - 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 + 3.85iT - 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 - 1.72iT - 83T^{2} \) |
| 89 | \( 1 + 0.589T + 89T^{2} \) |
| 97 | \( 1 - 4.45iT - 97T^{2} \) |
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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
−8.624346156354676010085809418369, −7.960274210008877322524798093484, −7.27882199568314052100985633290, −6.66653959995937796614818526723, −5.73945566625616285793144325119, −5.05853591577794764116373099917, −3.87733288632451208000334568842, −3.69021523723422618785678283523, −2.48917265180884561962651326322, −1.43551642322668975015247826991,
0.079459881287751585695375181001, 1.11349030404002871351056000504, 2.52427049317184096873356979138, 3.33363818439993116339223807291, 4.21804694304485466117013686813, 4.81791056779015936054301843743, 5.61190621192352242282134511915, 6.69626087349110517498475482052, 7.16544440141501295101010413059, 7.988233873730582806091322341189
