| L(s) = 1 | + (−0.277 + 0.584i)2-s + (2.04 + 1.88i)3-s + (0.999 + 1.22i)4-s + (−1.19 − 1.89i)5-s + (−1.66 + 0.672i)6-s + (−3.65 − 1.05i)7-s + (−2.25 + 0.554i)8-s + (0.380 + 4.71i)9-s + (1.43 − 0.173i)10-s + (3.68 + 4.32i)11-s + (−0.265 + 4.38i)12-s + (−2.22 + 2.83i)13-s + (1.63 − 1.84i)14-s + (1.12 − 6.11i)15-s + (−0.332 + 1.62i)16-s + (−1.33 − 2.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.196 + 0.413i)2-s + (1.18 + 1.08i)3-s + (0.499 + 0.612i)4-s + (−0.534 − 0.845i)5-s + (−0.681 + 0.274i)6-s + (−1.38 − 0.400i)7-s + (−0.795 + 0.196i)8-s + (0.126 + 1.57i)9-s + (0.454 − 0.0549i)10-s + (1.10 + 1.30i)11-s + (−0.0766 + 1.26i)12-s + (−0.617 + 0.786i)13-s + (0.436 − 0.492i)14-s + (0.290 − 1.57i)15-s + (−0.0830 + 0.407i)16-s + (−0.323 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.218572 + 1.52487i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218572 + 1.52487i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.19 + 1.89i)T \) |
| 13 | \( 1 + (2.22 - 2.83i)T \) |
| good | 2 | \( 1 + (0.277 - 0.584i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (-2.04 - 1.88i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (3.65 + 1.05i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (-3.68 - 4.32i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (1.33 + 2.42i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.185 + 0.0496i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (1.28 - 4.78i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.59 - 1.70i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-0.296 - 0.232i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (1.72 + 2.30i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (7.03 + 6.49i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-1.28 - 9.03i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (1.13 - 0.429i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (-4.62 + 7.65i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (-10.4 - 6.91i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-6.29 + 6.55i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-7.43 + 9.10i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (2.57 - 11.4i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (-6.76 + 9.80i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (3.47 - 1.31i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-7.31 - 13.9i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (-6.74 - 1.80i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (2.82 - 0.941i)T + (77.5 - 58.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05524984019472978972078930633, −9.435618170145682765908791389351, −9.105374464532022341902720515105, −8.168806333629900980664656875569, −7.17370338781296510065244827607, −6.70260614343321777155495588595, −4.95490486521498482718831859189, −3.95241634548554544013451513822, −3.52752419905989490856340764368, −2.22988715289846705167941738589,
0.65391125924669003235083114109, 2.28500717529105568481802755802, 3.00964181026068338839767578380, 3.61822475263549751279522329528, 5.94388561373881615856539263946, 6.56059264018123791032964887999, 7.03515237338241964687484231939, 8.294914462607826678347291951617, 8.825638037508523206093436946405, 9.871730976860608423469901340138
