| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s + 3·7-s − 8-s + 9-s − 10-s + 3·11-s + 2·12-s − 3·14-s + 2·15-s + 16-s − 2·17-s − 18-s + 2·19-s + 20-s + 6·21-s − 3·22-s − 8·23-s − 2·24-s + 25-s − 4·27-s + 3·28-s − 29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.577·12-s − 0.801·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s + 1.30·21-s − 0.639·22-s − 1.66·23-s − 0.408·24-s + 1/5·25-s − 0.769·27-s + 0.566·28-s − 0.185·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.85446207913149, −14.57048833394879, −14.10294366681140, −13.65102216643120, −13.18226929132561, −12.28062788331929, −11.75742371402658, −11.45553807058251, −10.71945826038191, −10.17953475186831, −9.560709876810498, −9.196695301917563, −8.557964033533381, −8.304826037013706, −7.686048521374208, −7.227048370744293, −6.434308909710036, −5.924739777032021, −5.149610800520353, −4.452519661266333, −3.717890713633589, −3.176227044234058, −2.246941015005402, −1.831303841422675, −1.331168576801413, 0,
1.331168576801413, 1.831303841422675, 2.246941015005402, 3.176227044234058, 3.717890713633589, 4.452519661266333, 5.149610800520353, 5.924739777032021, 6.434308909710036, 7.227048370744293, 7.686048521374208, 8.304826037013706, 8.557964033533381, 9.196695301917563, 9.560709876810498, 10.17953475186831, 10.71945826038191, 11.45553807058251, 11.75742371402658, 12.28062788331929, 13.18226929132561, 13.65102216643120, 14.10294366681140, 14.57048833394879, 14.85446207913149
