L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 3·11-s + 16-s − 2·17-s + 2·19-s + 20-s − 3·22-s + 23-s + 25-s + 5·29-s − 5·31-s + 32-s − 2·34-s + 3·37-s + 2·38-s + 40-s − 2·41-s − 9·43-s − 3·44-s + 46-s − 7·47-s − 7·49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.223·20-s − 0.639·22-s + 0.208·23-s + 1/5·25-s + 0.928·29-s − 0.898·31-s + 0.176·32-s − 0.342·34-s + 0.493·37-s + 0.324·38-s + 0.158·40-s − 0.312·41-s − 1.37·43-s − 0.452·44-s + 0.147·46-s − 1.02·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−16.16728388550889, −15.77979076405160, −15.07672000927166, −14.69580422523551, −13.93665884899272, −13.59374566510005, −12.93717147691319, −12.64632062107015, −11.86802908125662, −11.23993716767465, −10.80457923982219, −10.08572237203308, −9.637134243512886, −8.854880435072660, −8.126067040669623, −7.620263843232911, −6.807148015898287, −6.322659586768158, −5.634366209968207, −4.884617004547820, −4.640355438004227, −3.445675352577367, −3.008372474818991, −2.144454871153959, −1.393302660507237, 0,
1.393302660507237, 2.144454871153959, 3.008372474818991, 3.445675352577367, 4.640355438004227, 4.884617004547820, 5.634366209968207, 6.322659586768158, 6.807148015898287, 7.620263843232911, 8.126067040669623, 8.854880435072660, 9.637134243512886, 10.08572237203308, 10.80457923982219, 11.23993716767465, 11.86802908125662, 12.64632062107015, 12.93717147691319, 13.59374566510005, 13.93665884899272, 14.69580422523551, 15.07672000927166, 15.77979076405160, 16.16728388550889