| L(s) = 1 | − i·3-s + 0.648i·7-s − 9-s − 3.52·11-s − 1.35i·13-s + 6.87i·17-s + 19-s + 0.648·21-s − 5.46i·23-s + i·27-s − 5.52·29-s − 3.46·31-s + 3.52i·33-s + 0.0558i·37-s − 1.35·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.244i·7-s − 0.333·9-s − 1.06·11-s − 0.374i·13-s + 1.66i·17-s + 0.229·19-s + 0.141·21-s − 1.14i·23-s + 0.192i·27-s − 1.02·29-s − 0.622·31-s + 0.613i·33-s + 0.00917i·37-s − 0.216·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.523412332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.523412332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 0.648iT - 7T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 + 1.35iT - 13T^{2} \) |
| 17 | \( 1 - 6.87iT - 17T^{2} \) |
| 23 | \( 1 + 5.46iT - 23T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 0.0558iT - 37T^{2} \) |
| 41 | \( 1 - 9.52T + 41T^{2} \) |
| 43 | \( 1 + 7.69iT - 43T^{2} \) |
| 47 | \( 1 + 1.46iT - 47T^{2} \) |
| 53 | \( 1 - 13.2iT - 53T^{2} \) |
| 59 | \( 1 - 9.64T + 59T^{2} \) |
| 61 | \( 1 + 0.172T + 61T^{2} \) |
| 67 | \( 1 + 5.40iT - 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 6.34iT - 73T^{2} \) |
| 79 | \( 1 + 4.34T + 79T^{2} \) |
| 83 | \( 1 - 4.17iT - 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 + 2.64iT - 97T^{2} \) |
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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
−7.985309845626934566851507729014, −7.49577232748622677393995778735, −6.67052119875665825245852942833, −5.78088142040840874949606493910, −5.50958797067266204554985116466, −4.38005114435518950181336955669, −3.56287568496010836692297703465, −2.55126941085165675300421880034, −1.92179247036934687920652130782, −0.62588112750439738957519416075,
0.64927631806626242206423101939, 2.09674015552643546535285387791, 2.93735105833201849296368313429, 3.72921039810134132515393790486, 4.58218293818972653342017129375, 5.32654124920012394626521663334, 5.74711344382745520630935633101, 6.97942500199082809486570520322, 7.41995534509683620994910894215, 8.120752604210173331679769235163
