| L(s) = 1 | + (−2.68 − 0.874i)2-s + (−3.20 + 2.32i)3-s + (6.47 + 4.70i)4-s + (13.3 + 41.1i)5-s + (10.6 − 3.45i)6-s + (−7.80 + 10.7i)7-s + (−13.3 − 18.3i)8-s + (−20.1 + 62.1i)9-s − 122. i·10-s + (90.5 − 80.2i)11-s − 31.6·12-s + (−170. − 55.3i)13-s + (30.3 − 22.0i)14-s + (−138. − 100. i)15-s + (19.7 + 60.8i)16-s + (167. − 54.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 − 0.218i)2-s + (−0.355 + 0.258i)3-s + (0.404 + 0.293i)4-s + (0.535 + 1.64i)5-s + (0.295 − 0.0960i)6-s + (−0.159 + 0.219i)7-s + (−0.207 − 0.286i)8-s + (−0.249 + 0.767i)9-s − 1.22i·10-s + (0.748 − 0.662i)11-s − 0.219·12-s + (−1.00 − 0.327i)13-s + (0.154 − 0.112i)14-s + (−0.616 − 0.447i)15-s + (0.0772 + 0.237i)16-s + (0.580 − 0.188i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.677688 + 0.532002i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.677688 + 0.532002i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.68 + 0.874i)T \) |
| 11 | \( 1 + (-90.5 + 80.2i)T \) |
| good | 3 | \( 1 + (3.20 - 2.32i)T + (25.0 - 77.0i)T^{2} \) |
| 5 | \( 1 + (-13.3 - 41.1i)T + (-505. + 367. i)T^{2} \) |
| 7 | \( 1 + (7.80 - 10.7i)T + (-741. - 2.28e3i)T^{2} \) |
| 13 | \( 1 + (170. + 55.3i)T + (2.31e4 + 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-167. + 54.5i)T + (6.75e4 - 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-213. - 293. i)T + (-4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 - 564.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-505. + 695. i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 31 | \( 1 + (-276. + 849. i)T + (-7.47e5 - 5.42e5i)T^{2} \) |
| 37 | \( 1 + (164. + 119. i)T + (5.79e5 + 1.78e6i)T^{2} \) |
| 41 | \( 1 + (-1.54e3 - 2.11e3i)T + (-8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 + 2.60e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (1.25e3 - 908. i)T + (1.50e6 - 4.64e6i)T^{2} \) |
| 53 | \( 1 + (454. - 1.39e3i)T + (-6.38e6 - 4.63e6i)T^{2} \) |
| 59 | \( 1 + (-3.97e3 - 2.88e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (1.50e3 - 490. i)T + (1.12e7 - 8.13e6i)T^{2} \) |
| 67 | \( 1 + 3.00e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (2.62e3 + 8.08e3i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (-2.67e3 + 3.68e3i)T + (-8.77e6 - 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-2.00e3 - 653. i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (8.99e3 - 2.92e3i)T + (3.83e7 - 2.78e7i)T^{2} \) |
| 89 | \( 1 - 1.45e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-2.09e3 + 6.45e3i)T + (-7.16e7 - 5.20e7i)T^{2} \) |
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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
−17.54989266809395385673600636680, −16.57896840978946032552390504184, −14.98453519228970671732019677842, −13.87413277190241534084946318402, −11.75640312463564832903470816247, −10.67301444015640888700524560414, −9.692575721330587162300232410033, −7.57592986410147997887842231772, −6.00858708751557785474339884755, −2.81377302853264878004155944041,
1.04828963252608992778989220748, 5.10576407421415761283526258435, 6.90312270683465682746878318231, 8.875186065167477119342846526381, 9.708684524386178596923536102737, 11.87603926285664943252529573171, 12.77587628379006138057960765901, 14.53552394898692383310192625906, 16.19506573840168845841471738485, 17.21472663759446669476201235457
