| L(s) = 1 | + (−0.222 − 0.974i)2-s + (2.38 − 0.736i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (−1.25 − 2.16i)6-s + (0.287 − 0.497i)7-s + (0.623 + 0.781i)8-s + (2.68 − 1.83i)9-s + (−0.988 − 0.149i)10-s + (4.11 + 1.98i)11-s + (−1.83 + 1.70i)12-s + (0.587 − 0.0885i)13-s + (−0.548 − 0.169i)14-s + (0.186 − 2.49i)15-s + (0.623 − 0.781i)16-s + (−1.07 − 2.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.157 − 0.689i)2-s + (1.37 − 0.425i)3-s + (−0.450 + 0.216i)4-s + (0.163 − 0.416i)5-s + (−0.510 − 0.883i)6-s + (0.108 − 0.187i)7-s + (0.220 + 0.276i)8-s + (0.895 − 0.610i)9-s + (−0.312 − 0.0471i)10-s + (1.24 + 0.598i)11-s + (−0.529 + 0.490i)12-s + (0.162 − 0.0245i)13-s + (−0.146 − 0.0452i)14-s + (0.0482 − 0.643i)15-s + (0.155 − 0.195i)16-s + (−0.260 − 0.664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48882 - 1.33651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48882 - 1.33651i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.365 + 0.930i)T \) |
| 43 | \( 1 + (-4.21 - 5.02i)T \) |
| good | 3 | \( 1 + (-2.38 + 0.736i)T + (2.47 - 1.68i)T^{2} \) |
| 7 | \( 1 + (-0.287 + 0.497i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.11 - 1.98i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-0.587 + 0.0885i)T + (12.4 - 3.83i)T^{2} \) |
| 17 | \( 1 + (1.07 + 2.73i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (5.59 + 3.81i)T + (6.94 + 17.6i)T^{2} \) |
| 23 | \( 1 + (-0.344 - 4.60i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (4.24 + 1.30i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-1.54 + 1.43i)T + (2.31 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-4.26 - 7.38i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.10 + 4.85i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (8.34 - 4.02i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (8.68 + 1.30i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-5.49 + 6.89i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-2.24 - 2.07i)T + (4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (4.68 + 3.19i)T + (24.4 + 62.3i)T^{2} \) |
| 71 | \( 1 + (1.03 - 13.8i)T + (-70.2 - 10.5i)T^{2} \) |
| 73 | \( 1 + (-7.03 + 1.06i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-3.09 + 5.35i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.07 + 0.947i)T + (68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (-2.42 + 0.747i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + (-2.00 - 0.967i)T + (60.4 + 75.8i)T^{2} \) |
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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
−11.04810169539153656823140543746, −9.592847274680462958170030429772, −9.292575601112888001691013858934, −8.416652106307645433450841442224, −7.53975710094400698642825889449, −6.48506046522005031867489052122, −4.70385825941718699613415397473, −3.75776879797790634555731273676, −2.50854033864147278619520617195, −1.45225664974017292179906803038,
2.02815992588935333896841998014, 3.53118204706792726748803249615, 4.26609606114340345337541311903, 5.94387200363209223879502086287, 6.71636023134540171668354980499, 7.997283439899987844054003057268, 8.639169715825049198557190322968, 9.228650863674419466548441126065, 10.20562266413583789114949334062, 11.10037037710867410376269745983
