L(s) = 1 | − 1.95i·3-s + (0.479 − 2.18i)5-s − 2.28i·7-s − 0.840·9-s − 1.12·11-s − 5.95i·13-s + (−4.27 − 0.940i)15-s + 5.80i·17-s − 4.08·19-s − 4.47·21-s − i·23-s + (−4.53 − 2.09i)25-s − 4.23i·27-s + 0.408·29-s + 3.19·31-s + ⋯ |
L(s) = 1 | − 1.13i·3-s + (0.214 − 0.976i)5-s − 0.863i·7-s − 0.280·9-s − 0.338·11-s − 1.65i·13-s + (−1.10 − 0.242i)15-s + 1.40i·17-s − 0.936·19-s − 0.976·21-s − 0.208i·23-s + (−0.907 − 0.419i)25-s − 0.814i·27-s + 0.0758·29-s + 0.573·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.420443796\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.420443796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.479 + 2.18i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + 1.95iT - 3T^{2} \) |
| 7 | \( 1 + 2.28iT - 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 + 5.95iT - 13T^{2} \) |
| 17 | \( 1 - 5.80iT - 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 29 | \( 1 - 0.408T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 9.80iT - 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 - 7.75iT - 43T^{2} \) |
| 47 | \( 1 - 6.40iT - 47T^{2} \) |
| 53 | \( 1 - 6.73iT - 53T^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 + 0.283iT - 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 9.61iT - 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 + 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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
−8.554989634683303710952969310844, −7.86138124472733723838449056651, −7.57852445648555430477828610495, −6.28964074875278967996998402891, −5.88214651072731595959637212619, −4.70462682363974742598218718957, −3.89054495126938818068238641342, −2.50832698312760097287876785195, −1.39776402200889714222576843490, −0.53046410276135328497254604951,
2.05634196327415386491053317800, 2.86860946908592228367776784242, 3.91709286893344988738954271814, 4.72520866053954903162723754897, 5.52351354884817464765989288068, 6.58154337164540802573051399299, 7.07182369603570402573173533914, 8.302863920604641545441532968283, 9.172684767431059888862369204956, 9.626840171611874520235483444872