| L(s) = 1 | − 0.895i·2-s − i·3-s + 1.19·4-s − 0.895·6-s + 5.08i·7-s − 2.86i·8-s − 9-s + 2.64·11-s − 1.19i·12-s − 2.13i·13-s + 4.55·14-s − 0.167·16-s + 7.75i·17-s + 0.895i·18-s − 3.08·19-s + ⋯ |
| L(s) = 1 | − 0.633i·2-s − 0.577i·3-s + 0.599·4-s − 0.365·6-s + 1.92i·7-s − 1.01i·8-s − 0.333·9-s + 0.796·11-s − 0.345i·12-s − 0.591i·13-s + 1.21·14-s − 0.0419·16-s + 1.87i·17-s + 0.211i·18-s − 0.708·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.057726630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.057726630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.895iT - 2T^{2} \) |
| 7 | \( 1 - 5.08iT - 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 2.13iT - 13T^{2} \) |
| 17 | \( 1 - 7.75iT - 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 - 6.14iT - 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 + 0.0157iT - 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 3.81iT - 43T^{2} \) |
| 47 | \( 1 + 0.897iT - 47T^{2} \) |
| 53 | \( 1 - 9.26iT - 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 6.38T + 61T^{2} \) |
| 67 | \( 1 + 5.54iT - 67T^{2} \) |
| 71 | \( 1 + 0.0828T + 71T^{2} \) |
| 73 | \( 1 - 9.92iT - 73T^{2} \) |
| 79 | \( 1 + 5.30T + 79T^{2} \) |
| 83 | \( 1 + 0.723iT - 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 2.22iT - 97T^{2} \) |
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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
−9.175816692722648386734770360425, −8.510364890171109950598220811498, −7.79389481917761455471655469745, −6.64377102856585456593600183665, −6.06378205424855706857095997717, −5.50645625225775613321555162537, −3.97224977126145557330962804142, −3.02353661191302675055176108522, −2.15287852290791361851634041136, −1.43570709059061607989397882244,
0.76222906680940094898208044128, 2.31228279729569464605981999470, 3.54100557270458811390431597837, 4.39423611442234790846038316243, 5.04307390969517269766659615835, 6.38528699952590511788510421251, 6.86351598996935854817397899342, 7.40478129094573689926705029580, 8.349093683595193307233162159155, 9.193708646590931831398864359826
