| L(s) = 1 | − 3-s − 3.21i·5-s + 9-s + 1.36i·11-s + 2.93i·13-s + 3.21i·15-s − 6.91i·17-s + 7.35·19-s + 3.62i·23-s − 5.33·25-s − 27-s − 1.11·29-s + 8.70·31-s − 1.36i·33-s + 7.63·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.43i·5-s + 0.333·9-s + 0.412i·11-s + 0.812i·13-s + 0.830i·15-s − 1.67i·17-s + 1.68·19-s + 0.756i·23-s − 1.06·25-s − 0.192·27-s − 0.206·29-s + 1.56·31-s − 0.237i·33-s + 1.25·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.428693020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.428693020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.21iT - 5T^{2} \) |
| 11 | \( 1 - 1.36iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 6.91iT - 17T^{2} \) |
| 19 | \( 1 - 7.35T + 19T^{2} \) |
| 23 | \( 1 - 3.62iT - 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 - 0.833iT - 41T^{2} \) |
| 43 | \( 1 + 4.82iT - 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 + 4.57T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 - 6.49iT - 73T^{2} \) |
| 79 | \( 1 + 7.79iT - 79T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−9.139831335388443415717846174245, −7.902084156574564866324032739743, −7.38135666618645783879623071015, −6.41675322920327462087211142426, −5.42033726726524614802426606114, −4.89093436370137783432636053124, −4.29470592535421573507095367966, −2.97268586995458093004866658628, −1.55995617169335676175586798294, −0.64793286991988529143590192103,
1.10734586686064141873998906684, 2.63334102111600563974029850009, 3.30267428893147304543407570478, 4.28929399568925543005014295598, 5.44399006882325520829601389817, 6.19230068808898812919222328838, 6.61567379064356849634582659195, 7.71559656093709973831648275032, 8.064403232276560093206424747515, 9.343533067346510282931300645443
