| L(s) = 1 | + 1.27i·2-s − 1.16·3-s + 0.370·4-s − 1.81i·5-s − 1.49i·6-s + i·7-s + 3.02i·8-s − 1.63·9-s + 2.31·10-s − 2.77i·11-s − 0.432·12-s − 1.27·14-s + 2.11i·15-s − 3.12·16-s − 2.74·17-s − 2.08i·18-s + ⋯ |
| L(s) = 1 | + 0.902i·2-s − 0.674·3-s + 0.185·4-s − 0.811i·5-s − 0.608i·6-s + 0.377i·7-s + 1.06i·8-s − 0.545·9-s + 0.732·10-s − 0.837i·11-s − 0.124·12-s − 0.341·14-s + 0.547i·15-s − 0.780·16-s − 0.665·17-s − 0.492i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4566775778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4566775778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.27iT - 2T^{2} \) |
| 3 | \( 1 + 1.16T + 3T^{2} \) |
| 5 | \( 1 + 1.81iT - 5T^{2} \) |
| 11 | \( 1 + 2.77iT - 11T^{2} \) |
| 17 | \( 1 + 2.74T + 17T^{2} \) |
| 19 | \( 1 - 5.86iT - 19T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 - 2.06iT - 31T^{2} \) |
| 37 | \( 1 - 1.74iT - 37T^{2} \) |
| 41 | \( 1 - 6.36iT - 41T^{2} \) |
| 43 | \( 1 + 9.10T + 43T^{2} \) |
| 47 | \( 1 - 6.65iT - 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 3.07iT - 59T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 - 5.01iT - 67T^{2} \) |
| 71 | \( 1 + 2.71iT - 71T^{2} \) |
| 73 | \( 1 + 7.67iT - 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 7.97iT - 83T^{2} \) |
| 89 | \( 1 + 16.0iT - 89T^{2} \) |
| 97 | \( 1 - 14.2iT - 97T^{2} \) |
| 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
−10.24411793915256858933694380259, −9.099359099771088591896346800592, −8.314949322760783356457165430090, −7.945606177927601789902510318575, −6.58027844230262304347941702076, −6.01688876479814848090402774784, −5.45374843634660543764990980023, −4.58037374089167439529142465686, −3.12109686135065070249719811037, −1.68954950825539643068392092515,
0.19604297392388980089224172554, 1.94486429756595630177129316879, 2.80735080674473259056041298352, 3.86590677982701377828533947159, 4.88140516557462665352157359669, 6.10070079253308890186058105957, 6.81796532050196452201712539670, 7.37414200563384107499475527581, 8.681871956762871151924764167240, 9.738059282871232277982444858246
