| L(s) = 1 | + (−2.66 − 0.822i)2-s + (−6.33 + 16.1i)3-s + (−20.0 − 13.6i)4-s + (−58.2 + 8.78i)5-s + (30.1 − 37.8i)6-s + (54.7 − 117. i)7-s + (97.7 + 122. i)8-s + (−42.5 − 39.4i)9-s + (162. + 24.5i)10-s + (260. − 242. i)11-s + (347. − 236. i)12-s + (210. + 922. i)13-s + (−242. + 268. i)14-s + (227. − 997. i)15-s + (123. + 313. i)16-s + (−98.0 − 1.30e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.471 − 0.145i)2-s + (−0.406 + 1.03i)3-s + (−0.625 − 0.426i)4-s + (−1.04 + 0.157i)5-s + (0.342 − 0.429i)6-s + (0.422 − 0.906i)7-s + (0.540 + 0.677i)8-s + (−0.175 − 0.162i)9-s + (0.514 + 0.0775i)10-s + (0.650 − 0.603i)11-s + (0.695 − 0.474i)12-s + (0.345 + 1.51i)13-s + (−0.330 + 0.365i)14-s + (0.261 − 1.14i)15-s + (0.120 + 0.306i)16-s + (−0.0822 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.726961 - 0.197337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.726961 - 0.197337i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-54.7 + 117. i)T \) |
| good | 2 | \( 1 + (2.66 + 0.822i)T + (26.4 + 18.0i)T^{2} \) |
| 3 | \( 1 + (6.33 - 16.1i)T + (-178. - 165. i)T^{2} \) |
| 5 | \( 1 + (58.2 - 8.78i)T + (2.98e3 - 921. i)T^{2} \) |
| 11 | \( 1 + (-260. + 242. i)T + (1.20e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-210. - 922. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (98.0 + 1.30e3i)T + (-1.40e6 + 2.11e5i)T^{2} \) |
| 19 | \( 1 + (-1.28e3 + 2.22e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-250. + 3.33e3i)T + (-6.36e6 - 9.59e5i)T^{2} \) |
| 29 | \( 1 + (1.35e3 + 652. i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-4.22e3 - 7.31e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-6.62e3 + 4.51e3i)T + (2.53e7 - 6.45e7i)T^{2} \) |
| 41 | \( 1 + (-5.96e3 - 7.48e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-1.21e4 + 1.51e4i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (8.53e3 + 2.63e3i)T + (1.89e8 + 1.29e8i)T^{2} \) |
| 53 | \( 1 + (8.33e3 + 5.67e3i)T + (1.52e8 + 3.89e8i)T^{2} \) |
| 59 | \( 1 + (-3.38e4 - 5.09e3i)T + (6.83e8 + 2.10e8i)T^{2} \) |
| 61 | \( 1 + (-3.12e4 + 2.12e4i)T + (3.08e8 - 7.86e8i)T^{2} \) |
| 67 | \( 1 + (-5.99e3 - 1.03e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (2.54e4 - 1.22e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-1.60e4 + 4.94e3i)T + (1.71e9 - 1.16e9i)T^{2} \) |
| 79 | \( 1 + (1.05e3 - 1.83e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-527. + 2.31e3i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (2.46e4 + 2.28e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + 5.46e4T + 8.58e9T^{2} \) |
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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
−14.48120944388329396391514228319, −13.70860628245024155025399555250, −11.41911845844096618251046915070, −11.08596616255472108200795679330, −9.726563382767073613044246522687, −8.700459103607074489524774651622, −7.07840241398927089080039370167, −4.80584965578451218410575226711, −4.06375062691144418181316498039, −0.64674278431714245869780236556,
1.12323568683603715547392117071, 3.87569031319886506742727797674, 5.83891676365139591946334870213, 7.72505793930102224977244698328, 8.061287150844355167535510031306, 9.675874838053314810380746978909, 11.58597020340334997734451242530, 12.38602439602490240262732907508, 13.11592053883833335724704581876, 14.86035017182414357557119267948
