| L(s) = 1 | + (−6.01 − 11.8i)2-s + (−4.30 − 27.1i)3-s + (−65.6 + 90.3i)4-s + (−122. − 22.2i)5-s + (−295. + 214. i)6-s + (251. − 251. i)7-s + (623. + 98.7i)8-s + (−26.8 + 8.71i)9-s + (476. + 1.58e3i)10-s + (−349. + 1.07e3i)11-s + (2.73e3 + 1.39e3i)12-s + (643. − 1.26e3i)13-s + (−4.48e3 − 1.45e3i)14-s + (−76.4 + 3.43e3i)15-s + (−377. − 1.16e3i)16-s + (−428. + 2.70e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.752 − 1.47i)2-s + (−0.159 − 1.00i)3-s + (−1.02 + 1.41i)4-s + (−0.983 − 0.178i)5-s + (−1.36 + 0.992i)6-s + (0.734 − 0.734i)7-s + (1.21 + 0.192i)8-s + (−0.0367 + 0.0119i)9-s + (0.476 + 1.58i)10-s + (−0.262 + 0.807i)11-s + (1.58 + 0.806i)12-s + (0.292 − 0.574i)13-s + (−1.63 − 0.531i)14-s + (−0.0226 + 1.01i)15-s + (−0.0920 − 0.283i)16-s + (−0.0871 + 0.550i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.323655 + 0.290662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.323655 + 0.290662i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (122. + 22.2i)T \) |
| good | 2 | \( 1 + (6.01 + 11.8i)T + (-37.6 + 51.7i)T^{2} \) |
| 3 | \( 1 + (4.30 + 27.1i)T + (-693. + 225. i)T^{2} \) |
| 7 | \( 1 + (-251. + 251. i)T - 1.17e5iT^{2} \) |
| 11 | \( 1 + (349. - 1.07e3i)T + (-1.43e6 - 1.04e6i)T^{2} \) |
| 13 | \( 1 + (-643. + 1.26e3i)T + (-2.83e6 - 3.90e6i)T^{2} \) |
| 17 | \( 1 + (428. - 2.70e3i)T + (-2.29e7 - 7.45e6i)T^{2} \) |
| 19 | \( 1 + (6.08e3 + 8.36e3i)T + (-1.45e7 + 4.47e7i)T^{2} \) |
| 23 | \( 1 + (2.10e4 - 1.07e4i)T + (8.70e7 - 1.19e8i)T^{2} \) |
| 29 | \( 1 + (1.41e4 - 1.95e4i)T + (-1.83e8 - 5.65e8i)T^{2} \) |
| 31 | \( 1 + (-1.48e4 + 1.07e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (-1.58e3 - 806. i)T + (1.50e9 + 2.07e9i)T^{2} \) |
| 41 | \( 1 + (9.07e3 + 2.79e4i)T + (-3.84e9 + 2.79e9i)T^{2} \) |
| 43 | \( 1 + (8.49e4 + 8.49e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 + (-1.66e5 + 2.63e4i)T + (1.02e10 - 3.33e9i)T^{2} \) |
| 53 | \( 1 + (3.55e4 + 2.24e5i)T + (-2.10e10 + 6.84e9i)T^{2} \) |
| 59 | \( 1 + (-7.61e4 + 2.47e4i)T + (3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-2.06e4 + 6.36e4i)T + (-4.16e10 - 3.02e10i)T^{2} \) |
| 67 | \( 1 + (2.69e4 - 1.69e5i)T + (-8.60e10 - 2.79e10i)T^{2} \) |
| 71 | \( 1 + (4.24e5 + 3.08e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (3.25e5 - 1.65e5i)T + (8.89e10 - 1.22e11i)T^{2} \) |
| 79 | \( 1 + (1.66e5 - 2.29e5i)T + (-7.51e10 - 2.31e11i)T^{2} \) |
| 83 | \( 1 + (-3.69e5 - 5.84e4i)T + (3.10e11 + 1.01e11i)T^{2} \) |
| 89 | \( 1 + (5.22e5 + 1.69e5i)T + (4.02e11 + 2.92e11i)T^{2} \) |
| 97 | \( 1 + (-3.97e5 + 6.29e4i)T + (7.92e11 - 2.57e11i)T^{2} \) |
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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
−15.34869410602139696815876988996, −13.31244099117166701686184131947, −12.38816267317912332717518496395, −11.41832661529342535276609510863, −10.30223414827713116611763722049, −8.399833891175067278410881790344, −7.35679920555829729683649568926, −4.06830388174191123855573662377, −1.78568078207415937704552408555, −0.34006931710787722883712055221,
4.39972323624768478787198790296, 6.00991987965276022695339435113, 7.86997006794704052893285363788, 8.760137096455012203340502581406, 10.32731267804101603961842352498, 11.76592761939501244898180047870, 14.30340144921124308167556847746, 15.22099337737485226516665160740, 16.05210162363444447950772066891, 16.66414103954683153383837583546
