| L(s) = 1 | + (1.32 − 1.11i)3-s + 5-s − 1.85i·7-s + (0.492 − 2.95i)9-s + 5.93·11-s − 1.57·13-s + (1.32 − 1.11i)15-s + 4.36·17-s − 0.298i·19-s + (−2.07 − 2.44i)21-s + (−1.25 + 4.62i)23-s + 25-s + (−2.66 − 4.46i)27-s + 6.89i·29-s − 9.74·31-s + ⋯ |
| L(s) = 1 | + (0.762 − 0.646i)3-s + 0.447·5-s − 0.699i·7-s + (0.164 − 0.986i)9-s + 1.78·11-s − 0.436·13-s + (0.341 − 0.289i)15-s + 1.05·17-s − 0.0684i·19-s + (−0.452 − 0.533i)21-s + (−0.262 + 0.964i)23-s + 0.200·25-s + (−0.512 − 0.858i)27-s + 1.27i·29-s − 1.75·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.596536815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.596536815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.32 + 1.11i)T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (1.25 - 4.62i)T \) |
| good | 7 | \( 1 + 1.85iT - 7T^{2} \) |
| 11 | \( 1 - 5.93T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 + 0.298iT - 19T^{2} \) |
| 29 | \( 1 - 6.89iT - 29T^{2} \) |
| 31 | \( 1 + 9.74T + 31T^{2} \) |
| 37 | \( 1 + 7.74iT - 37T^{2} \) |
| 41 | \( 1 + 7.49iT - 41T^{2} \) |
| 43 | \( 1 - 4.96iT - 43T^{2} \) |
| 47 | \( 1 + 4.73iT - 47T^{2} \) |
| 53 | \( 1 - 2.15T + 53T^{2} \) |
| 59 | \( 1 + 3.27iT - 59T^{2} \) |
| 61 | \( 1 - 12.6iT - 61T^{2} \) |
| 67 | \( 1 + 13.5iT - 67T^{2} \) |
| 71 | \( 1 - 1.09iT - 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 + 5.00iT - 79T^{2} \) |
| 83 | \( 1 - 5.01T + 83T^{2} \) |
| 89 | \( 1 - 8.95T + 89T^{2} \) |
| 97 | \( 1 - 11.2iT - 97T^{2} \) |
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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
−9.191258360428458129300078405089, −8.904709559703308503917581394404, −7.50119678210524241838643105519, −7.25798704529090725675607422343, −6.31155044203120479455608320759, −5.38247423402469064272502098622, −3.91394231876621275106226011107, −3.45166539037712165659134798907, −1.96074177149332572165560323968, −1.10951048006255772023543905717,
1.58736644403111952728719586083, 2.64803663634045470161291540364, 3.67290770067987829322516511695, 4.51179599246171167000842143949, 5.55227484483251265002236558231, 6.36227986995741023260187750679, 7.39764348156098885277835383849, 8.365836082226616731287234497113, 9.013442356013818540969264781374, 9.659347937198623308866384072741
