| L(s) = 1 | − 1.23i·3-s − i·5-s + 3.23·7-s + 1.47·9-s − 2i·11-s + 4.47i·13-s − 1.23·15-s + 4.47·17-s + 4.47i·19-s − 4.00i·21-s + 4.76·23-s − 25-s − 5.52i·27-s + 2i·29-s + 6.47·31-s + ⋯ |
| L(s) = 1 | − 0.713i·3-s − 0.447i·5-s + 1.22·7-s + 0.490·9-s − 0.603i·11-s + 1.24i·13-s − 0.319·15-s + 1.08·17-s + 1.02i·19-s − 0.872i·21-s + 0.993·23-s − 0.200·25-s − 1.06i·27-s + 0.371i·29-s + 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.124165015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124165015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.23iT - 3T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47iT - 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6.94iT - 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 7.70iT - 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 - 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 8.47iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 7.70iT - 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 3.70iT - 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−9.554027262986394771446854042015, −8.386517380225325543173944353204, −8.148847576557974407246196464284, −7.12710693846070451145161831632, −6.39549452819894005964596905258, −5.24971666641931078667979733510, −4.57637837066157715345068957727, −3.42555364820719516974219742284, −1.81908960562838514009757382627, −1.21747628466182784883573378349,
1.25242770333003207017693364841, 2.68971168765809430516447262763, 3.70998350958847169357695749324, 4.90465943014338747054474285055, 5.12932362238935979701801348881, 6.52114040423523938238352679120, 7.50321057633018936740893674413, 8.022046031839539330568827744009, 9.054567580012149841845600751266, 9.981776233612549904907853223251
