| L(s) = 1 | + (1.57 + 0.511i)2-s + (1.16 + 1.59i)3-s + (0.596 + 0.433i)4-s + (1.09 + 1.95i)5-s + (1.00 + 3.10i)6-s + (−1.31 + 1.81i)7-s + (−1.22 − 1.68i)8-s + (−0.278 + 0.855i)9-s + (0.718 + 3.62i)10-s + 1.45i·12-s + (3.51 + 1.14i)13-s + (−3.00 + 2.18i)14-s + (−1.85 + 4.00i)15-s + (−1.52 − 4.68i)16-s + (2.11 − 0.687i)17-s + (−0.875 + 1.20i)18-s + ⋯ |
| L(s) = 1 | + (1.11 + 0.361i)2-s + (0.670 + 0.922i)3-s + (0.298 + 0.216i)4-s + (0.487 + 0.872i)5-s + (0.412 + 1.26i)6-s + (−0.498 + 0.685i)7-s + (−0.434 − 0.597i)8-s + (−0.0926 + 0.285i)9-s + (0.227 + 1.14i)10-s + 0.420i·12-s + (0.975 + 0.317i)13-s + (−0.802 + 0.583i)14-s + (−0.478 + 1.03i)15-s + (−0.380 − 1.17i)16-s + (0.513 − 0.166i)17-s + (−0.206 + 0.283i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98207 + 2.35940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98207 + 2.35940i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.09 - 1.95i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.57 - 0.511i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.16 - 1.59i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (1.31 - 1.81i)T + (-2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-3.51 - 1.14i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.11 + 0.687i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (4.27 - 3.10i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 3.85iT - 23T^{2} \) |
| 29 | \( 1 + (0.152 + 0.110i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.212 + 0.653i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.52 - 2.09i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.40 + 4.65i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 8.41iT - 43T^{2} \) |
| 47 | \( 1 + (7.06 + 9.71i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-12.0 - 3.91i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (0.278 + 0.202i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.535 + 1.64i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 0.650iT - 67T^{2} \) |
| 71 | \( 1 + (-1.43 - 4.42i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.20 + 7.16i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.23 - 6.88i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.02 + 0.983i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 9.92T + 89T^{2} \) |
| 97 | \( 1 + (2.15 + 0.700i)T + (78.4 + 57.0i)T^{2} \) |
| 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
−10.67583053313450288653965088817, −10.02152145060781931650975249488, −9.217187013103529843432740187822, −8.487838541904557863271356775521, −6.89958063503148430777337935491, −6.18658051825211254771159884635, −5.44070085208926885259999390939, −4.09891470460764829040516769566, −3.51185912742449193612763325097, −2.49565299517664100862310801815,
1.30136988302279891189027460927, 2.57967521742358309286089004646, 3.67740823723875564017819010622, 4.66061734197760132426530415723, 5.76416787978614633853918091396, 6.61295536312690491173332124700, 7.86254407503062121589099921043, 8.535193381869956504300074352528, 9.413017271814010286074868667498, 10.59354442767788312114501638501
