L(s) = 1 | − 2.23·5-s + 2.64i·7-s − 3·9-s + 5.29·17-s + 4.47i·19-s + 5.00·25-s − 5.91i·35-s − 11.8·37-s − 11.8i·43-s + 6.70·45-s + 5.29i·47-s − 7.00·49-s − 11.8·53-s − 13.4i·59-s − 4.47·61-s + ⋯ |
L(s) = 1 | − 0.999·5-s + 0.999i·7-s − 9-s + 1.28·17-s + 1.02i·19-s + 1.00·25-s − 0.999i·35-s − 1.94·37-s − 1.80i·43-s + 0.999·45-s + 0.771i·47-s − 49-s − 1.62·53-s − 1.74i·59-s − 0.572·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1392484646\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1392484646\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 4.47iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 11.8iT - 43T^{2} \) |
| 47 | \( 1 - 5.29iT - 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 13.4iT - 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 11.8iT - 67T^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
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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
−8.585287568363781894551930416126, −8.110670989145508145313287961543, −7.37557518492435525760481178441, −6.28592632541514961732786515447, −5.56000541984338865457110041340, −4.87239464518316193629598058683, −3.54205577643517214603907287313, −3.11636700067053529757904703687, −1.77763662732572721587579988683, −0.05306235113101741117486665452,
1.16681786651119622616479779470, 2.91279056352169359755294907878, 3.50036850623202572377120306897, 4.48391111922263059517538684135, 5.24329581810857029720994326537, 6.29692520096888007496319150813, 7.19814134261492146789340993192, 7.71824577259872785155013711472, 8.476953002104302987242592760620, 9.177652008694077114482179019094