| L(s) = 1 | + (1.20 − 0.438i)2-s + (−0.520 + 2.95i)3-s + (−4.87 + 4.08i)4-s + (1.07 + 0.900i)5-s + (0.667 + 3.78i)6-s + (13.8 + 24.0i)7-s + (−9.19 + 15.9i)8-s + (−8.45 − 3.07i)9-s + (1.68 + 0.614i)10-s + (−1.17 + 2.03i)11-s + (−9.53 − 16.5i)12-s + (−2.75 − 15.6i)13-s + (27.2 + 22.8i)14-s + (−3.22 + 2.70i)15-s + (4.73 − 26.8i)16-s + (111. − 40.5i)17-s + ⋯ |
| L(s) = 1 | + (0.425 − 0.154i)2-s + (−0.100 + 0.568i)3-s + (−0.608 + 0.510i)4-s + (0.0960 + 0.0805i)5-s + (0.0454 + 0.257i)6-s + (0.749 + 1.29i)7-s + (−0.406 + 0.704i)8-s + (−0.313 − 0.114i)9-s + (0.0533 + 0.0194i)10-s + (−0.0322 + 0.0558i)11-s + (−0.229 − 0.397i)12-s + (−0.0588 − 0.333i)13-s + (0.520 + 0.436i)14-s + (−0.0554 + 0.0465i)15-s + (0.0740 − 0.419i)16-s + (1.58 − 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.09763 + 0.973102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09763 + 0.973102i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.520 - 2.95i)T \) |
| 19 | \( 1 + (71.0 - 42.5i)T \) |
| good | 2 | \( 1 + (-1.20 + 0.438i)T + (6.12 - 5.14i)T^{2} \) |
| 5 | \( 1 + (-1.07 - 0.900i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-13.8 - 24.0i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (1.17 - 2.03i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (2.75 + 15.6i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-111. + 40.5i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-41.0 + 34.4i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-166. - 60.4i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-67.5 - 117. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 121.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (10.8 - 61.5i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-1.87 - 1.57i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-542. - 197. i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + (183. - 153. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-137. + 49.9i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (224. - 188. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-739. - 269. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (643. + 539. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-148. + 839. i)T + (-3.65e5 - 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-36.8 + 209. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (702. + 1.21e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (236. + 1.33e3i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (1.53e3 - 557. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
| 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
−14.73615963701380102841246741159, −14.09705973440943897905961598092, −12.41717761690506722617533634283, −11.93819003752051305832480682357, −10.36963772553880806495927006105, −8.962360415550354210786247128142, −8.073767635074232233004600410580, −5.71664852696719174465310930928, −4.67387840976837699680451035822, −2.89858428393485998999131503281,
1.08472195085739053813336984329, 4.06569163806089868955170042081, 5.45751720606893121762420177024, 6.95968616049081193436640876434, 8.276022583785318810604100134976, 9.881870528836628477193172017266, 10.99194821524583960881991503699, 12.47380656554960657014530744845, 13.59283630737550749579440834190, 14.17362118906944060256547124291
