| L(s) = 1 | + (1.08 − 0.910i)2-s + (2.23 + 1.13i)3-s + (0.342 − 1.97i)4-s + (−1.70 − 1.44i)5-s + (3.45 − 0.801i)6-s + 1.03·7-s + (−1.42 − 2.44i)8-s + (1.93 + 2.66i)9-s + (−3.16 − 0.0101i)10-s + (3.45 − 0.546i)11-s + (3.01 − 4.01i)12-s + (−0.657 + 4.15i)13-s + (1.12 − 0.943i)14-s + (−2.16 − 5.17i)15-s + (−3.76 − 1.35i)16-s + (−1.05 − 0.341i)17-s + ⋯ |
| L(s) = 1 | + (0.765 − 0.643i)2-s + (1.29 + 0.657i)3-s + (0.171 − 0.985i)4-s + (−0.763 − 0.646i)5-s + (1.41 − 0.327i)6-s + 0.391·7-s + (−0.502 − 0.864i)8-s + (0.644 + 0.887i)9-s + (−0.999 − 0.00321i)10-s + (1.04 − 0.164i)11-s + (0.868 − 1.15i)12-s + (−0.182 + 1.15i)13-s + (0.299 − 0.252i)14-s + (−0.560 − 1.33i)15-s + (−0.941 − 0.337i)16-s + (−0.255 − 0.0828i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.51051 - 1.14258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.51051 - 1.14258i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.08 + 0.910i)T \) |
| 5 | \( 1 + (1.70 + 1.44i)T \) |
| good | 3 | \( 1 + (-2.23 - 1.13i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + (-3.45 + 0.546i)T + (10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (0.657 - 4.15i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (1.05 + 0.341i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.953 - 0.486i)T + (11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (3.71 + 2.69i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.525 + 1.03i)T + (-17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (1.38 - 4.27i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (5.85 + 0.926i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-6.02 - 8.29i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.20 - 8.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.50 + 0.813i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.21 + 2.37i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (1.26 - 7.96i)T + (-56.1 - 18.2i)T^{2} \) |
| 61 | \( 1 + (-0.0773 - 0.488i)T + (-58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (4.97 + 9.76i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (11.3 - 3.69i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (12.2 + 8.89i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (1.97 + 6.08i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.69 - 3.32i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-8.73 + 12.0i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.61 - 0.524i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.38316014509595186037740194547, −10.22930420332102969843647306222, −9.130501864957510048880040923106, −8.852554287609932003321193942033, −7.58264917492505821692653336995, −6.24819793357092229112598748358, −4.51920334807581120244132136949, −4.26206063991565892796224905779, −3.15479317657588207004542276896, −1.71229636069030736923602117908,
2.27026078443841968797427612641, 3.39170230246519387824259209822, 4.18181777568874568293136193677, 5.78737167577082050090760015624, 7.02473324003928233336921760302, 7.55427785626779266370781988100, 8.298863294170164906177207858974, 9.103608850571089052139608898989, 10.65363587247420868035371921993, 11.74746677056608611351039817657
