| L(s) = 1 | + (−3.66 − 2.66i)2-s + (7.06 + 9.72i)3-s + (1.41 + 4.34i)4-s − 44.3·5-s − 54.5i·6-s + (17.8 + 54.9i)7-s + (−16.0 + 49.2i)8-s + (−19.6 + 60.4i)9-s + (162. + 118. i)10-s + (−130. + 42.3i)11-s + (−32.3 + 44.4i)12-s + (23.2 + 31.9i)13-s + (80.9 − 249. i)14-s + (−313. − 431. i)15-s + (249. − 181. i)16-s + (285. + 92.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.917 − 0.666i)2-s + (0.785 + 1.08i)3-s + (0.0882 + 0.271i)4-s − 1.77·5-s − 1.51i·6-s + (0.364 + 1.12i)7-s + (−0.250 + 0.770i)8-s + (−0.242 + 0.746i)9-s + (1.62 + 1.18i)10-s + (−1.07 + 0.349i)11-s + (−0.224 + 0.308i)12-s + (0.137 + 0.189i)13-s + (0.412 − 1.27i)14-s + (−1.39 − 1.91i)15-s + (0.974 − 0.707i)16-s + (0.988 + 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.278985 + 0.431158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.278985 + 0.431158i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-661. - 697. i)T \) |
| good | 2 | \( 1 + (3.66 + 2.66i)T + (4.94 + 15.2i)T^{2} \) |
| 3 | \( 1 + (-7.06 - 9.72i)T + (-25.0 + 77.0i)T^{2} \) |
| 5 | \( 1 + 44.3T + 625T^{2} \) |
| 7 | \( 1 + (-17.8 - 54.9i)T + (-1.94e3 + 1.41e3i)T^{2} \) |
| 11 | \( 1 + (130. - 42.3i)T + (1.18e4 - 8.60e3i)T^{2} \) |
| 13 | \( 1 + (-23.2 - 31.9i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-285. - 92.8i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (426. + 309. i)T + (4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 + (241. + 78.5i)T + (2.26e5 + 1.64e5i)T^{2} \) |
| 29 | \( 1 + (114. - 157. i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 37 | \( 1 - 972. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-208. - 151. i)T + (8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 + (1.70e3 - 2.34e3i)T + (-1.05e6 - 3.25e6i)T^{2} \) |
| 47 | \( 1 + (2.01e3 - 1.46e3i)T + (1.50e6 - 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-1.64e3 - 535. i)T + (6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (-1.70e3 + 1.23e3i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 - 4.44e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.10e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (421. - 1.29e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (-4.57e3 + 1.48e3i)T + (2.29e7 - 1.66e7i)T^{2} \) |
| 79 | \( 1 + (1.99e3 + 649. i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (488. - 672. i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 + (-3.62e3 + 1.17e3i)T + (5.07e7 - 3.68e7i)T^{2} \) |
| 97 | \( 1 + (2.92e3 + 9.01e3i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−16.24599303405132207764714974388, −15.22150486309398973929914024332, −14.81121561780965504633971341112, −12.30602367616515408965396768999, −11.21951159901854144230748879624, −10.09747146115563845529401779663, −8.662668936084070517639877858046, −8.126343285489941827336002387340, −4.72900246297458642514685501030, −2.91467796914792910882035922142,
0.46866457158664630946288294189, 3.70876973446125502769015174062, 7.08408589256843363112420704289, 7.998586245417716542200762142126, 8.177597187758791052562106547211, 10.49808113114910375098720222841, 12.18058809876088603013861536389, 13.31643169559792382753517524600, 14.76574876575459256196606056268, 15.95539244526263381792656639727
