| L(s) = 1 | − 2.68i·3-s + 3.18i·7-s − 4.18·9-s − 0.681i·13-s + 1.18i·17-s − 19-s + 8.55·21-s − 2.17i·23-s + 3.18i·27-s − 2.81·29-s + 6.37·31-s − 7.87i·37-s − 1.82·39-s + 0.983·41-s + 1.36i·43-s + ⋯ |
| L(s) = 1 | − 1.54i·3-s + 1.20i·7-s − 1.39·9-s − 0.188i·13-s + 0.288i·17-s − 0.229·19-s + 1.86·21-s − 0.453i·23-s + 0.613i·27-s − 0.521·29-s + 1.14·31-s − 1.29i·37-s − 0.292·39-s + 0.153·41-s + 0.207i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.215897408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215897408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.68iT - 3T^{2} \) |
| 7 | \( 1 - 3.18iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.681iT - 13T^{2} \) |
| 17 | \( 1 - 1.18iT - 17T^{2} \) |
| 23 | \( 1 + 2.17iT - 23T^{2} \) |
| 29 | \( 1 + 2.81T + 29T^{2} \) |
| 31 | \( 1 - 6.37T + 31T^{2} \) |
| 37 | \( 1 + 7.87iT - 37T^{2} \) |
| 41 | \( 1 - 0.983T + 41T^{2} \) |
| 43 | \( 1 - 1.36iT - 43T^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 + 1.69iT - 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 7.36T + 61T^{2} \) |
| 67 | \( 1 + 7.02iT - 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 5.53iT - 73T^{2} \) |
| 79 | \( 1 + 5.36T + 79T^{2} \) |
| 83 | \( 1 - 2.37iT - 83T^{2} \) |
| 89 | \( 1 + 3.01T + 89T^{2} \) |
| 97 | \( 1 + 4.88iT - 97T^{2} \) |
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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
−8.221816399255326661475320467941, −7.40892776960227057789642078789, −6.74615932107751762551100395850, −5.98658062194261384487469943741, −5.56164227805018773157424575510, −4.43685415299385165283372326625, −3.15188261646090351558425730707, −2.33068654746527686165677783329, −1.68738148471675665015981328022, −0.37173644232868418529930824677,
1.19105542465414532992862668459, 2.77814777577673496579588190349, 3.57495112561867474205829030122, 4.34110647764074058205566265299, 4.71402143406213401837962130829, 5.67991062778095768005943445613, 6.55555989541343719961267021208, 7.41607900878032059559042503366, 8.143234769137201269677742368950, 9.042962221810729308856415128900
