| L(s) = 1 | + (−1.26 − 0.642i)2-s + (−2.50 − 0.396i)3-s + (1.17 + 1.61i)4-s + (2.89 + 2.10i)6-s + (1.84 − 1.84i)7-s + (−0.442 − 2.79i)8-s + (−2.45 − 0.797i)9-s + (0.224 + 0.692i)11-s + (−2.30 − 4.51i)12-s + (−10.9 + 5.57i)13-s + (−3.51 + 1.14i)14-s + (−1.23 + 3.80i)16-s + (20.9 − 3.31i)17-s + (2.57 + 2.57i)18-s + (−4.52 + 6.22i)19-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.321i)2-s + (−0.834 − 0.132i)3-s + (0.293 + 0.404i)4-s + (0.483 + 0.351i)6-s + (0.263 − 0.263i)7-s + (−0.0553 − 0.349i)8-s + (−0.272 − 0.0885i)9-s + (0.0204 + 0.0629i)11-s + (−0.191 − 0.376i)12-s + (−0.842 + 0.429i)13-s + (−0.250 + 0.0815i)14-s + (−0.0772 + 0.237i)16-s + (1.23 − 0.194i)17-s + (0.143 + 0.143i)18-s + (−0.237 + 0.327i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.495554 + 0.301523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.495554 + 0.301523i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 + 0.642i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (2.50 + 0.396i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-1.84 + 1.84i)T - 49iT^{2} \) |
| 11 | \( 1 + (-0.224 - 0.692i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (10.9 - 5.57i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-20.9 + 3.31i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (4.52 - 6.22i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (18.0 - 35.3i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-28.3 - 38.9i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-31.6 - 22.9i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (8.18 + 16.0i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (2.02 - 6.24i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (15.0 + 15.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (13.1 - 83.0i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-64.5 - 10.2i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (74.4 + 24.2i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (20.0 + 61.8i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (61.2 - 9.69i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (43.2 - 31.4i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-15.4 + 30.3i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (13.7 + 18.9i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-8.10 - 51.1i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-142. + 46.4i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (9.98 - 63.0i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−12.13674135307849674919451836874, −11.01891259803205306498569635156, −10.20117018063709680604613560027, −9.272432845515808374874497462515, −8.047039549278638462256878175196, −7.13453121041894979530866071966, −5.97251534941800797918185603902, −4.81861081373990316394692304403, −3.16014927803747304543914064238, −1.31612372381928540216006868435,
0.45012297378133186263167267847, 2.55119806775269298222179129332, 4.64161689847870353111831778535, 5.67443129943576719715001407737, 6.52081340771020706495093254957, 7.87692883683676331893539214372, 8.575812725938161878976154569371, 10.06994689651070167207104708355, 10.40581414752114304951992907388, 11.84555423873070798269105783758
