| L(s) = 1 | − 2·5-s + 4·11-s + (−3 − 2i)13-s + 6·17-s − 6·19-s − 25-s + 6i·29-s + 6·37-s + 2i·41-s − 12i·43-s − 8i·47-s + 7·49-s + 6i·53-s − 8·55-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + (−0.832 − 0.554i)13-s + 1.45·17-s − 1.37·19-s − 0.200·25-s + 1.11i·29-s + 0.986·37-s + 0.312i·41-s − 1.82i·43-s − 1.16i·47-s + 49-s + 0.824i·53-s − 1.07·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.183039516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183039516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 4iT - 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.393920770199289480267053353172, −7.51045477216154741283237030315, −7.09522272047666056082519411103, −6.11891328359257354704132726465, −5.35935691289634428079140618089, −4.36263764546326121484929839330, −3.77614109645117648439817455852, −2.93103350413379264729290238770, −1.68447006369564314962793817433, −0.41731324462198897463502655657,
1.01200000737079765270822036503, 2.22901930167024324517131399439, 3.31580729280092770838851682676, 4.21171094241068599702197745484, 4.55068114152757839241900205072, 5.87584362936218668652861887067, 6.40879934491785292857139281944, 7.38587368524324033774351611285, 7.82368740047487608994891099157, 8.609293857833091431151835799591
