L(s) = 1 | − 2.63i·2-s + (2.12 − 2.12i)3-s + 1.07·4-s + 9.61i·5-s + (−5.58 − 5.58i)6-s + (6.96 − 6.96i)7-s − 23.8i·8-s − 8.99i·9-s + 25.2·10-s + (20.2 − 20.2i)11-s + (2.28 − 2.28i)12-s + (28.8 − 28.8i)13-s + (−18.3 − 18.3i)14-s + (20.3 + 20.3i)15-s − 54.2·16-s + (4.20 + 4.20i)17-s + ⋯ |
L(s) = 1 | − 0.930i·2-s + (0.408 − 0.408i)3-s + 0.134·4-s + 0.859i·5-s + (−0.379 − 0.379i)6-s + (0.376 − 0.376i)7-s − 1.05i·8-s − 0.333i·9-s + 0.799·10-s + (0.555 − 0.555i)11-s + (0.0550 − 0.0550i)12-s + (0.616 − 0.616i)13-s + (−0.349 − 0.349i)14-s + (0.350 + 0.350i)15-s − 0.847·16-s + (0.0600 + 0.0600i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.39531 - 1.65213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39531 - 1.65213i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.12 + 2.12i)T \) |
| 41 | \( 1 + (151. - 214. i)T \) |
good | 2 | \( 1 + 2.63iT - 8T^{2} \) |
| 5 | \( 1 - 9.61iT - 125T^{2} \) |
| 7 | \( 1 + (-6.96 + 6.96i)T - 343iT^{2} \) |
| 11 | \( 1 + (-20.2 + 20.2i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (-28.8 + 28.8i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (-4.20 - 4.20i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + (0.701 + 0.701i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (169. - 169. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 78.4T + 5.06e4T^{2} \) |
| 43 | \( 1 + 59.3iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-77.4 - 77.4i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (175. - 175. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 - 499.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 297. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + (-530. - 530. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + (-129. + 129. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + 318. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (396. - 396. i)T - 4.93e5iT^{2} \) |
| 83 | \( 1 - 8.78T + 5.71e5T^{2} \) |
| 89 | \( 1 + (608. - 608. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + (1.05e3 + 1.05e3i)T + 9.12e5iT^{2} \) |
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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
−12.54614557805016666717527883320, −11.46267831530899132477412711285, −10.79886366130684892441600557241, −9.830399394200909347174341383622, −8.371825051520023918616176183219, −7.16691017802720868628834076240, −6.15183932355890929240696965012, −3.84481228809896890655302734671, −2.80783323818071945800199479507, −1.25931017057559438469528527700,
1.98464356431036170422148100479, 4.19566964842940100800770117531, 5.38670620172108564196782032776, 6.60675209211150130551465634213, 7.986202904253592152164070456845, 8.701163238099842699649777998085, 9.770846681510945128094761209269, 11.31946178947158861992750769130, 12.14095401413536464938535511409, 13.55469374514927141751855439606