| L(s) = 1 | + (−1.69 − 1.15i)2-s + (1.28 + 1.19i)3-s + (0.811 + 2.06i)4-s + (0.653 − 0.201i)5-s + (−0.799 − 3.50i)6-s + (−2.34 − 1.21i)7-s + (0.100 − 0.441i)8-s + (0.00460 + 0.0614i)9-s + (−1.34 − 0.414i)10-s + (0.294 − 3.92i)11-s + (−1.41 + 3.61i)12-s + (−0.900 + 0.433i)13-s + (2.58 + 4.78i)14-s + (1.07 + 0.519i)15-s + (2.57 − 2.38i)16-s + (−5.35 + 0.807i)17-s + ⋯ |
| L(s) = 1 | + (−1.20 − 0.818i)2-s + (0.740 + 0.687i)3-s + (0.405 + 1.03i)4-s + (0.292 − 0.0902i)5-s + (−0.326 − 1.43i)6-s + (−0.888 − 0.459i)7-s + (0.0355 − 0.155i)8-s + (0.00153 + 0.0204i)9-s + (−0.424 − 0.131i)10-s + (0.0887 − 1.18i)11-s + (−0.409 + 1.04i)12-s + (−0.249 + 0.120i)13-s + (0.689 + 1.27i)14-s + (0.278 + 0.134i)15-s + (0.643 − 0.597i)16-s + (−1.29 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.123815 - 0.509705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123815 - 0.509705i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.34 + 1.21i)T \) |
| 13 | \( 1 + (0.900 - 0.433i)T \) |
| good | 2 | \( 1 + (1.69 + 1.15i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.28 - 1.19i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.653 + 0.201i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.294 + 3.92i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (5.35 - 0.807i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (0.297 + 0.515i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.45 + 0.520i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-1.81 - 2.27i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-0.831 + 1.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.23 + 8.24i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.08 + 9.15i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.0187 - 0.0821i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (8.34 + 5.68i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (4.69 + 11.9i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-1.08 - 0.333i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (4.00 - 10.1i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-1.57 + 2.72i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.70 + 5.89i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (12.8 - 8.77i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-3.08 - 5.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.64 - 2.23i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.572 - 7.63i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 6.03T + 97T^{2} \) |
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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
−10.02668303681967293101489991515, −9.382241177694389367436395157898, −8.849121395906467870884746394597, −8.117256866984844460919674003000, −6.84054726695070266705452288707, −5.74430404688264917725250321383, −4.07855192384505732898647061328, −3.24519183687027887727572732753, −2.18103807817236985640986863552, −0.36420783366255801434205782811,
1.78693548698538647369138098574, 2.84565684196327935039661009439, 4.56688313583767420189450385844, 6.22728830061646356140707598430, 6.64292339161488991076122744966, 7.65389304370113339723840610363, 8.176704129641653623397040291548, 9.189341673854298450961110830303, 9.676744129907982907794447508792, 10.42351078689932749340138627961
