| L(s) = 1 | + (1.61 − 2.22i)2-s + (12.9 − 4.22i)3-s + (7.55 + 23.2i)4-s + (30.9 + 46.5i)5-s + (11.6 − 35.7i)6-s − 221. i·7-s + (147. + 47.9i)8-s + (−45.5 + 33.0i)9-s + (153. + 6.42i)10-s + (196. + 142. i)11-s + (196. + 270. i)12-s + (−354. − 488. i)13-s + (−491. − 357. i)14-s + (598. + 474. i)15-s + (−287. + 209. i)16-s + (−1.26e3 − 410. i)17-s + ⋯ |
| L(s) = 1 | + (0.285 − 0.393i)2-s + (0.833 − 0.270i)3-s + (0.236 + 0.726i)4-s + (0.553 + 0.832i)5-s + (0.131 − 0.405i)6-s − 1.70i·7-s + (0.815 + 0.264i)8-s + (−0.187 + 0.136i)9-s + (0.485 + 0.0203i)10-s + (0.489 + 0.355i)11-s + (0.393 + 0.541i)12-s + (−0.582 − 0.801i)13-s + (−0.670 − 0.486i)14-s + (0.686 + 0.544i)15-s + (−0.281 + 0.204i)16-s + (−1.05 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.27705 - 0.239163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.27705 - 0.239163i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-30.9 - 46.5i)T \) |
| good | 2 | \( 1 + (-1.61 + 2.22i)T + (-9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-12.9 + 4.22i)T + (196. - 142. i)T^{2} \) |
| 7 | \( 1 + 221. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-196. - 142. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (354. + 488. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.26e3 + 410. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (651. - 2.00e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.16e3 + 1.60e3i)T + (-1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (2.04e3 + 6.28e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-942. + 2.90e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-4.27e3 - 5.88e3i)T + (-2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-5.08e3 + 3.69e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.00e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (8.98e3 - 2.91e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-8.09e3 + 2.62e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.43e4 + 1.04e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.68e4 - 2.67e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.35e4 - 4.40e3i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-5.14e3 - 1.58e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.67e4 - 2.29e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.18e4 - 6.71e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-5.64e4 - 1.83e4i)T + (3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-2.03e4 - 1.48e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-7.68e3 + 2.49e3i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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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
−16.83428910656987832088358007105, −14.79415754586733996671440880213, −13.79452285895127272521090724889, −13.05208313150705329713409408234, −11.21044372594622398459884925048, −10.07609385417894935276611167565, −7.972780230684744572184613872585, −6.96343308070490923028786705356, −3.91704783251574729150023701133, −2.37461788677188235349262170540,
2.18540414398729055599482678942, 4.97368390796157735690666359252, 6.33391174914649102140711108060, 8.894140598940810551130433104434, 9.284424247947634927554803029534, 11.47154186469918944303146771276, 13.05648436276309732063832296205, 14.38824277556827104612333536770, 15.15916869514208414237170252800, 16.19065164903883834702646112089
