| L(s) = 1 | + (−0.212 + 0.977i)2-s + (0.800 − 0.599i)3-s + (−0.909 − 0.415i)4-s + (1.24 − 1.85i)5-s + (0.415 + 0.909i)6-s + (−0.283 + 3.96i)7-s + (0.599 − 0.800i)8-s + (0.281 − 0.959i)9-s + (1.55 + 1.60i)10-s + (3.02 − 4.69i)11-s + (−0.977 + 0.212i)12-s + (1.81 − 0.129i)13-s + (−3.81 − 1.12i)14-s + (−0.118 − 2.23i)15-s + (0.654 + 0.755i)16-s + (−0.817 + 2.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.690i)2-s + (0.462 − 0.345i)3-s + (−0.454 − 0.207i)4-s + (0.555 − 0.831i)5-s + (0.169 + 0.371i)6-s + (−0.107 + 1.49i)7-s + (0.211 − 0.283i)8-s + (0.0939 − 0.319i)9-s + (0.490 + 0.509i)10-s + (0.910 − 1.41i)11-s + (−0.282 + 0.0613i)12-s + (0.503 − 0.0360i)13-s + (−1.01 − 0.299i)14-s + (−0.0306 − 0.576i)15-s + (0.163 + 0.188i)16-s + (−0.198 + 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79351 + 0.119458i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79351 + 0.119458i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.212 - 0.977i)T \) |
| 3 | \( 1 + (-0.800 + 0.599i)T \) |
| 5 | \( 1 + (-1.24 + 1.85i)T \) |
| 23 | \( 1 + (3.84 - 2.86i)T \) |
| good | 7 | \( 1 + (0.283 - 3.96i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (-3.02 + 4.69i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.81 + 0.129i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (0.817 - 2.19i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.83 + 4.01i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-4.61 + 2.10i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.739 - 5.14i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.36 - 6.16i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-8.10 + 2.37i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (5.46 + 7.29i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-6.16 + 6.16i)T - 47iT^{2} \) |
| 53 | \( 1 + (-12.9 - 0.925i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-1.89 - 1.63i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.75 + 0.251i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (0.547 + 0.119i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (10.7 - 6.90i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (12.4 - 4.62i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (2.72 - 3.14i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (12.4 - 6.79i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (0.0853 - 0.593i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.72 + 0.942i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.15728821041929687340098375062, −9.123324592725867199352785998488, −8.691228504028932538021420677380, −8.322602124114486546583681315051, −6.81445246253907285685887753964, −5.91382433014173378200511915933, −5.50209798397757029097736999532, −4.03716921223473727635609875253, −2.64971761091002606517128281789, −1.18037345966483791973410627052,
1.43969842765168909715697653349, 2.68924542118944495066414785078, 3.93915662158308497218022256534, 4.39094561817643714670206868875, 6.09529424631332081525916877439, 7.13529609657694808528110738995, 7.73902181800279832059491797255, 9.097942880835641402116678625278, 9.918105017674265319462241080839, 10.20632944540528537364210222504
