L(s) = 1 | − 3i·3-s − 20i·7-s − 9·9-s − 56·11-s − 86i·13-s + 106i·17-s − 4·19-s − 60·21-s + 136i·23-s + 27i·27-s + 206·29-s − 152·31-s + 168i·33-s − 282i·37-s − 258·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.07i·7-s − 0.333·9-s − 1.53·11-s − 1.83i·13-s + 1.51i·17-s − 0.0482·19-s − 0.623·21-s + 1.23i·23-s + 0.192i·27-s + 1.31·29-s − 0.880·31-s + 0.886i·33-s − 1.25i·37-s − 1.05·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 20iT - 343T^{2} \) |
| 11 | \( 1 + 56T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 106iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 136iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 206T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152T + 2.97e4T^{2} \) |
| 37 | \( 1 + 282iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 246T + 6.89e4T^{2} \) |
| 43 | \( 1 - 412iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 40iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 126iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 56T + 2.05e5T^{2} \) |
| 61 | \( 1 + 2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 388iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 672T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.17e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 408T + 4.93e5T^{2} \) |
| 83 | \( 1 - 668iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 66T + 7.04e5T^{2} \) |
| 97 | \( 1 - 926iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02625051401963210467891274654, −8.420965240853881810552966229208, −7.86197692270884753898154690220, −7.21287775648466853315084680363, −5.93299351961544439853575582028, −5.20234415738391452486477451965, −3.77651573908943329558227462050, −2.74502401999040569148335862030, −1.23501937385760517758318041152, 0,
2.17341438147151103132788865928, 2.99618877926623599753681107198, 4.59538274183147697469467042266, 5.12500783878217281851563795407, 6.28594904743092998073887404083, 7.27567543719548659570757669931, 8.522105176246537065099730048452, 9.034437486085000684684456868724, 9.961330259366870301714752079040