| L(s) = 1 | − 3i·7-s − 11-s − 5.52i·13-s − 4.52i·17-s + 19-s − 8.52i·23-s + 2·29-s + 5.52·31-s − 8.52i·37-s + 0.520·41-s + 1.52i·43-s + 10.5i·47-s − 2·49-s + 2i·53-s + 10.5·59-s + ⋯ |
| L(s) = 1 | − 1.13i·7-s − 0.301·11-s − 1.53i·13-s − 1.09i·17-s + 0.229·19-s − 1.77i·23-s + 0.371·29-s + 0.991·31-s − 1.40i·37-s + 0.0813·41-s + 0.231i·43-s + 1.53i·47-s − 0.285·49-s + 0.274i·53-s + 1.36·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.702645216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.702645216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 13 | \( 1 + 5.52iT - 13T^{2} \) |
| 17 | \( 1 + 4.52iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 8.52iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 + 8.52iT - 37T^{2} \) |
| 41 | \( 1 - 0.520T + 41T^{2} \) |
| 43 | \( 1 - 1.52iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 15.5iT - 67T^{2} \) |
| 71 | \( 1 + 6.52T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 - 11.0iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14.0iT - 97T^{2} \) |
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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
−7.59277659194123218618914908567, −6.70509623125241510275585606597, −6.10654495670703948866752731856, −5.18339359050377180582197900955, −4.66356102131741425528100012508, −3.88862484243525947374620832133, −2.98827203260322351614480199681, −2.45247854847071643418254244601, −0.948321312291163334207701466377, −0.44732252933028528883935978831,
1.35627885894512844696235767803, 2.06158247462042068926037450949, 2.88253730930274350388828723805, 3.78485921277467481779337734420, 4.46895964118218847022530733027, 5.39723720953053975263404688689, 5.78159587717714064796536732136, 6.66118468881377280402125320564, 7.11870241261007946757939913993, 8.140492853408032434914222168705
