| L(s) = 1 | + (0.349 + 0.936i)2-s + (1.41 + 2.58i)3-s + (−0.755 + 0.654i)4-s + (0.600 − 2.15i)5-s + (−1.93 + 2.22i)6-s + (−3.44 + 2.57i)7-s + (−0.877 − 0.479i)8-s + (−3.08 + 4.79i)9-s + (2.22 − 0.190i)10-s + (2.62 − 1.19i)11-s + (−2.76 − 1.03i)12-s + (0.656 − 0.876i)13-s + (−3.61 − 2.32i)14-s + (6.42 − 1.49i)15-s + (0.142 − 0.989i)16-s + (6.87 − 0.491i)17-s + ⋯ |
| L(s) = 1 | + (0.247 + 0.662i)2-s + (0.816 + 1.49i)3-s + (−0.377 + 0.327i)4-s + (0.268 − 0.963i)5-s + (−0.788 + 0.909i)6-s + (−1.30 + 0.974i)7-s + (−0.310 − 0.169i)8-s + (−1.02 + 1.59i)9-s + (0.704 − 0.0601i)10-s + (0.789 − 0.360i)11-s + (−0.797 − 0.297i)12-s + (0.182 − 0.243i)13-s + (−0.967 − 0.621i)14-s + (1.65 − 0.384i)15-s + (0.0355 − 0.247i)16-s + (1.66 − 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.759167 + 1.41843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759167 + 1.41843i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.349 - 0.936i)T \) |
| 5 | \( 1 + (-0.600 + 2.15i)T \) |
| 23 | \( 1 + (-3.83 - 2.87i)T \) |
| good | 3 | \( 1 + (-1.41 - 2.58i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (3.44 - 2.57i)T + (1.97 - 6.71i)T^{2} \) |
| 11 | \( 1 + (-2.62 + 1.19i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.656 + 0.876i)T + (-3.66 - 12.4i)T^{2} \) |
| 17 | \( 1 + (-6.87 + 0.491i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (0.238 + 0.274i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (7.29 + 6.31i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-2.32 + 0.682i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (0.0397 + 0.182i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (1.27 - 0.818i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-7.35 + 4.01i)T + (23.2 - 36.1i)T^{2} \) |
| 47 | \( 1 + (2.00 - 2.00i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.72 + 3.64i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (7.97 - 1.14i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.0664 - 0.226i)T + (-51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (1.69 - 0.631i)T + (50.6 - 43.8i)T^{2} \) |
| 71 | \( 1 + (2.82 - 6.17i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (0.120 - 1.68i)T + (-72.2 - 10.3i)T^{2} \) |
| 79 | \( 1 + (2.16 + 15.0i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (9.93 - 2.16i)T + (75.4 - 34.4i)T^{2} \) |
| 89 | \( 1 + (-16.0 - 4.72i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (9.23 + 2.00i)T + (88.2 + 40.2i)T^{2} \) |
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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
−12.75205482815174091397481972494, −11.72178266377235812430983643444, −10.05346935308057190093497957186, −9.325976509928457361485176565016, −8.994131779329652579902777509546, −7.87548664220630749206137826659, −6.00326234924946449739492757551, −5.32520144535626421899110638070, −3.96840701718318539285359022061, −3.06983360297340670118812839885,
1.36788346937142646828424177635, 2.93963674790016289547754492075, 3.64758942525281448440369046013, 6.10364601042132430787433096196, 6.90433650441320270003226787310, 7.58490006789393034143116927722, 9.119637070743680284427665427183, 9.932781922367311355845053491131, 10.97614365744675576199679817499, 12.26234857873186264300173724137
