| L(s) = 1 | + (−3.14 − 0.970i)2-s + (−3.81 + 9.71i)3-s + (−17.4 − 11.9i)4-s + (67.5 − 10.1i)5-s + (21.4 − 26.8i)6-s + (−81.7 − 100. i)7-s + (109. + 136. i)8-s + (98.3 + 91.2i)9-s + (−222. − 33.5i)10-s + (−557. + 517. i)11-s + (182. − 124. i)12-s + (36.0 + 158. i)13-s + (159. + 395. i)14-s + (−158. + 695. i)15-s + (36.9 + 94.0i)16-s + (121. + 1.62e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.556 − 0.171i)2-s + (−0.244 + 0.622i)3-s + (−0.546 − 0.372i)4-s + (1.20 − 0.182i)5-s + (0.242 − 0.304i)6-s + (−0.630 − 0.776i)7-s + (0.602 + 0.755i)8-s + (0.404 + 0.375i)9-s + (−0.703 − 0.106i)10-s + (−1.39 + 1.28i)11-s + (0.365 − 0.249i)12-s + (0.0592 + 0.259i)13-s + (0.217 + 0.539i)14-s + (−0.182 + 0.797i)15-s + (0.0360 + 0.0918i)16-s + (0.102 + 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.514443 + 0.589381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.514443 + 0.589381i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (81.7 + 100. i)T \) |
| good | 2 | \( 1 + (3.14 + 0.970i)T + (26.4 + 18.0i)T^{2} \) |
| 3 | \( 1 + (3.81 - 9.71i)T + (-178. - 165. i)T^{2} \) |
| 5 | \( 1 + (-67.5 + 10.1i)T + (2.98e3 - 921. i)T^{2} \) |
| 11 | \( 1 + (557. - 517. i)T + (1.20e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-36.0 - 158. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-121. - 1.62e3i)T + (-1.40e6 + 2.11e5i)T^{2} \) |
| 19 | \( 1 + (-257. + 446. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (249. - 3.32e3i)T + (-6.36e6 - 9.59e5i)T^{2} \) |
| 29 | \( 1 + (-3.84e3 - 1.85e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (2.37e3 + 4.11e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-5.05e3 + 3.44e3i)T + (2.53e7 - 6.45e7i)T^{2} \) |
| 41 | \( 1 + (838. + 1.05e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (6.60e3 - 8.27e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-1.05e4 - 3.26e3i)T + (1.89e8 + 1.29e8i)T^{2} \) |
| 53 | \( 1 + (7.54e3 + 5.14e3i)T + (1.52e8 + 3.89e8i)T^{2} \) |
| 59 | \( 1 + (-3.76e4 - 5.67e3i)T + (6.83e8 + 2.10e8i)T^{2} \) |
| 61 | \( 1 + (3.39e4 - 2.31e4i)T + (3.08e8 - 7.86e8i)T^{2} \) |
| 67 | \( 1 + (8.15e3 + 1.41e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (6.54e4 - 3.14e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-2.75e4 + 8.48e3i)T + (1.71e9 - 1.16e9i)T^{2} \) |
| 79 | \( 1 + (-3.43e4 + 5.95e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.69e4 + 1.18e5i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (1.26e4 + 1.17e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + 3.92e4T + 8.58e9T^{2} \) |
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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
−14.96943182844140875200450828088, −13.48150024868228315647061300031, −13.03101427185707062267820450132, −10.63761672248036684763150263882, −10.09127392543231758581897740209, −9.433718471262146810911850764683, −7.60916704266531689506943921860, −5.66429766893269423619772840943, −4.48003395970039809881499922602, −1.75069479359874275268767536857,
0.50763853660104686257115576039, 2.81023160924011501068541998005, 5.44577083103592613124953780477, 6.66040248539698948732076528069, 8.204833937215682903007937327953, 9.406225108537649444179872994510, 10.35682541461703502849817690083, 12.25786636946508707411705794036, 13.21314530056168944541440464085, 13.85640395126304234719890184684
