| L(s) = 1 | + 0.470i·3-s + 2.71i·7-s + 2.77·9-s − 5.55·11-s − 2.02i·13-s − 3.77i·17-s − 19-s − 1.28·21-s + 5.77i·23-s + 2.71i·27-s + 5.66·29-s + 7.55·31-s − 2.61i·33-s − 3.75i·37-s + 0.954·39-s + ⋯ |
| L(s) = 1 | + 0.271i·3-s + 1.02i·7-s + 0.926·9-s − 1.67·11-s − 0.562i·13-s − 0.916i·17-s − 0.229·19-s − 0.279·21-s + 1.20i·23-s + 0.523i·27-s + 1.05·29-s + 1.35·31-s − 0.455i·33-s − 0.616i·37-s + 0.152·39-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
| \(L(1)\) |
\(\approx\) |
\(0.8948177051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8948177051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.470iT - 3T^{2} \) |
| 7 | \( 1 - 2.71iT - 7T^{2} \) |
| 11 | \( 1 + 5.55T + 11T^{2} \) |
| 13 | \( 1 + 2.02iT - 13T^{2} \) |
| 17 | \( 1 + 3.77iT - 17T^{2} \) |
| 23 | \( 1 - 5.77iT - 23T^{2} \) |
| 29 | \( 1 - 5.66T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 3.75iT - 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 - 9.43iT - 43T^{2} \) |
| 47 | \( 1 - 11.1iT - 47T^{2} \) |
| 53 | \( 1 - 8.85iT - 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 11.5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.45iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4.94iT - 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−8.836597046037778600421564544218, −7.920158719713411235101499469094, −7.60470985382915291578906609626, −6.53808952417902425738244094436, −5.71409813749702915123146125377, −5.00860081746807648424293948768, −4.49659202727815626737415738950, −3.04870458607400675070429381942, −2.69703934389160025444259154045, −1.38505100929604384259484377236,
0.25823291424071232560391514923, 1.51051979957863450251381268019, 2.47954697627113768009821596601, 3.55674168602339080079632327461, 4.49502115445240503263295553096, 4.92075589908933804379742365196, 6.14672898070967587722297320638, 6.84043099622044586997038794199, 7.34420713014736610772179935454, 8.281536264587853182533690338085
