L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s − 2·9-s − 10-s − 2·11-s − 12-s − 5·13-s + 14-s + 15-s + 16-s − 2·18-s + 5·19-s − 20-s − 21-s − 2·22-s + 2·23-s − 24-s + 25-s − 5·26-s + 5·27-s + 28-s + 29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s − 1.38·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.471·18-s + 1.14·19-s − 0.223·20-s − 0.218·21-s − 0.426·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.962·27-s + 0.188·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−15.92691444928443, −15.23258086736412, −14.82769123304444, −14.33704935313380, −13.80720649441815, −13.13562175366590, −12.55790958874302, −12.00625662085275, −11.63651361404859, −11.17578139300612, −10.54867925394655, −9.962847600935098, −9.295112663272763, −8.490178314309368, −7.827123830587909, −7.397398329729497, −6.768338921760130, −6.028675236629629, −5.240246884356200, −5.084944250368611, −4.419331150213048, −3.454277011870232, −2.844967236797420, −2.206043027806020, −1.010004793270361, 0,
1.010004793270361, 2.206043027806020, 2.844967236797420, 3.454277011870232, 4.419331150213048, 5.084944250368611, 5.240246884356200, 6.028675236629629, 6.768338921760130, 7.397398329729497, 7.827123830587909, 8.490178314309368, 9.295112663272763, 9.962847600935098, 10.54867925394655, 11.17578139300612, 11.63651361404859, 12.00625662085275, 12.55790958874302, 13.13562175366590, 13.80720649441815, 14.33704935313380, 14.82769123304444, 15.23258086736412, 15.92691444928443