L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s − 2·13-s + 4·14-s + 16-s − 6·17-s + 20-s + 25-s + 2·26-s − 4·28-s − 6·29-s − 8·31-s − 32-s + 6·34-s − 4·35-s − 2·37-s − 40-s − 6·41-s − 4·43-s + 9·49-s − 50-s − 2·52-s − 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.755·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.676·35-s − 0.328·37-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.141·50-s − 0.277·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.68871891855849, −15.13057775903490, −14.68518867848415, −13.87992790883157, −13.35802426434237, −12.86660415454204, −12.56665288933569, −11.77537623885046, −11.23026605287229, −10.59800939405654, −10.19182791619861, −9.520735329122987, −9.196600711554548, −8.839330674022020, −7.958790934135412, −7.327073450451754, −6.670242106999230, −6.524188486789435, −5.684073447136196, −5.146590038241415, −4.195465799735141, −3.500051894332702, −2.864515048081545, −2.165258349072296, −1.506294887114446, 0, 0,
1.506294887114446, 2.165258349072296, 2.864515048081545, 3.500051894332702, 4.195465799735141, 5.146590038241415, 5.684073447136196, 6.524188486789435, 6.670242106999230, 7.327073450451754, 7.958790934135412, 8.839330674022020, 9.196600711554548, 9.520735329122987, 10.19182791619861, 10.59800939405654, 11.23026605287229, 11.77537623885046, 12.56665288933569, 12.86660415454204, 13.35802426434237, 13.87992790883157, 14.68518867848415, 15.13057775903490, 15.68871891855849