| L(s) = 1 | + 4·7-s − 13-s + 2·17-s + 2·29-s + 4·31-s − 6·37-s + 6·41-s + 4·43-s + 4·47-s + 9·49-s − 10·53-s − 2·61-s + 8·67-s + 4·71-s + 6·73-s + 8·79-s − 8·83-s + 6·89-s − 4·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.277·13-s + 0.485·17-s + 0.371·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.583·47-s + 9/7·49-s − 1.37·53-s − 0.256·61-s + 0.977·67-s + 0.474·71-s + 0.702·73-s + 0.900·79-s − 0.878·83-s + 0.635·89-s − 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.242123710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.242123710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.44281960885523, −14.12357329439058, −13.86262234515068, −12.97653458414057, −12.47936789421998, −12.00148908009308, −11.48107623502752, −11.01594245651293, −10.51758304873910, −9.984646286642103, −9.305226682395713, −8.771553338136482, −8.187838473989369, −7.739328727030440, −7.343663026158838, −6.522441033867393, −5.983500604386605, −5.143476914571762, −4.949459568642040, −4.232177791552386, −3.600831332849887, −2.732950623329903, −2.105078505610335, −1.392639637954782, −0.6768436092840287,
0.6768436092840287, 1.392639637954782, 2.105078505610335, 2.732950623329903, 3.600831332849887, 4.232177791552386, 4.949459568642040, 5.143476914571762, 5.983500604386605, 6.522441033867393, 7.343663026158838, 7.739328727030440, 8.187838473989369, 8.771553338136482, 9.305226682395713, 9.984646286642103, 10.51758304873910, 11.01594245651293, 11.48107623502752, 12.00148908009308, 12.47936789421998, 12.97653458414057, 13.86262234515068, 14.12357329439058, 14.44281960885523
