| L(s) = 1 | + (−0.906 + 0.422i)2-s + (−0.0197 − 1.73i)3-s + (0.642 − 0.766i)4-s + (1.82 − 1.28i)5-s + (0.749 + 1.56i)6-s + (−2.23 + 0.599i)7-s + (−0.258 + 0.965i)8-s + (−2.99 + 0.0683i)9-s + (−1.11 + 1.93i)10-s + (−4.47 + 2.58i)11-s + (−1.33 − 1.09i)12-s + (−3.32 − 4.74i)13-s + (1.77 − 1.48i)14-s + (−2.26 − 3.13i)15-s + (−0.173 − 0.984i)16-s + (−3.71 + 1.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.640 + 0.298i)2-s + (−0.0113 − 0.999i)3-s + (0.321 − 0.383i)4-s + (0.817 − 0.576i)5-s + (0.306 + 0.637i)6-s + (−0.845 + 0.226i)7-s + (−0.0915 + 0.341i)8-s + (−0.999 + 0.0227i)9-s + (−0.351 + 0.613i)10-s + (−1.35 + 0.779i)11-s + (−0.386 − 0.317i)12-s + (−0.922 − 1.31i)13-s + (0.473 − 0.397i)14-s + (−0.585 − 0.810i)15-s + (−0.0434 − 0.246i)16-s + (−0.900 + 0.419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0166936 + 0.313828i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0166936 + 0.313828i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.906 - 0.422i)T \) |
| 3 | \( 1 + (0.0197 + 1.73i)T \) |
| 5 | \( 1 + (-1.82 + 1.28i)T \) |
| 19 | \( 1 + (-4.35 + 0.273i)T \) |
| good | 7 | \( 1 + (2.23 - 0.599i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (4.47 - 2.58i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.32 + 4.74i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (3.71 - 1.73i)T + (10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (-0.253 - 2.89i)T + (-22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (3.37 - 1.22i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.06 - 3.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.36 + 4.36i)T + 37iT^{2} \) |
| 41 | \( 1 + (-5.60 + 0.988i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.946 + 10.8i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.232 + 0.499i)T + (-30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (0.614 + 7.02i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (3.30 + 1.20i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (4.90 + 4.11i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-11.0 - 5.13i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-4.62 - 5.51i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (7.92 + 5.55i)T + (24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (0.848 - 0.149i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (16.6 - 4.46i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (1.20 - 6.84i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (3.47 + 7.44i)T + (-62.3 + 74.3i)T^{2} \) |
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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
−10.07991992003786192280878261173, −9.394126778233976451428445631222, −8.458382764529517238985336447105, −7.53262094780182298950522696676, −6.90885409819773047522108427826, −5.59949736485088813003335913344, −5.30467985107675924768063394954, −2.89140674671998468535283216980, −1.95344243579446260724588829529, −0.19672044249549570281188183428,
2.45209290466828035991471983903, 3.15071662514281414727911908424, 4.58441841464379036372873876719, 5.74730085544910474249740376934, 6.69195614870557220954360350847, 7.72198045106208597413631157284, 9.037382452110524087410899806797, 9.553945354764967362784544774535, 10.15769682794778392065941290781, 10.97837574449056079967394297701
