| L(s) = 1 | − i·2-s − 1.70i·3-s − 4-s + 1.22i·5-s − 1.70·6-s + 3.68·7-s + i·8-s + 0.0980·9-s + 1.22·10-s − 11-s + 1.70i·12-s − 6.29·13-s − 3.68i·14-s + 2.07·15-s + 16-s + 4.97·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.983i·3-s − 0.5·4-s + 0.545i·5-s − 0.695·6-s + 1.39·7-s + 0.353i·8-s + 0.0326·9-s + 0.386·10-s − 0.301·11-s + 0.491i·12-s − 1.74·13-s − 0.984i·14-s + 0.536·15-s + 0.250·16-s + 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743694606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743694606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| 53 | \( 1 + (-6.44 - 3.38i)T \) |
| good | 3 | \( 1 + 1.70iT - 3T^{2} \) |
| 5 | \( 1 - 1.22iT - 5T^{2} \) |
| 7 | \( 1 - 3.68T + 7T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 - 0.808iT - 19T^{2} \) |
| 23 | \( 1 + 6.17iT - 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 + 9.46iT - 31T^{2} \) |
| 37 | \( 1 - 9.37T + 37T^{2} \) |
| 41 | \( 1 + 4.27iT - 41T^{2} \) |
| 43 | \( 1 + 2.11T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 - 0.804iT - 61T^{2} \) |
| 67 | \( 1 - 11.4iT - 67T^{2} \) |
| 71 | \( 1 - 0.761iT - 71T^{2} \) |
| 73 | \( 1 + 9.51iT - 73T^{2} \) |
| 79 | \( 1 - 4.18iT - 79T^{2} \) |
| 83 | \( 1 + 6.93iT - 83T^{2} \) |
| 89 | \( 1 + 7.53T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.859225106731400896674315764669, −8.502743988313728742386666335117, −7.75202080184452147076041805140, −7.35371998988440547866884908170, −6.21166220684206299768859790578, −5.03297243942750312434478959739, −4.36998491511127851041580360427, −2.74946304388112963323966996017, −2.12231100352181614742207336112, −0.872894689796175454856685997697,
1.36882096222224009292922981248, 3.09407416997689066975088768505, 4.43581928296233182363982749638, 5.03334432831116364415441376962, 5.29260396272248130556319564274, 6.88207865530752757458748388515, 7.75224051425062238841633593234, 8.262166133220446207335052928793, 9.322657213887924139405241300359, 9.867191344317999285182046122218
