L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 2·11-s − 12-s − 4·13-s − 14-s + 16-s − 3·17-s + 18-s + 21-s + 2·22-s − 24-s − 4·26-s − 27-s − 28-s − 6·29-s − 8·31-s + 32-s − 2·33-s − 3·34-s + 36-s − 11·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.218·21-s + 0.426·22-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.348·33-s − 0.514·34-s + 1/6·36-s − 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.51466845529548, −12.34454737519852, −11.95929998491538, −11.34791874277799, −11.02022764563857, −10.59948992095448, −10.09725067445848, −9.443878108926148, −9.315652404807452, −8.652404085536158, −8.051497099318524, −7.404813807275197, −7.018075787086882, −6.786881689512721, −6.179104781785633, −5.669875818870397, −5.161919225431572, −4.906483929190479, −4.211787086444181, −3.676993012198466, −3.421071910556062, −2.564438762659587, −1.981808336726995, −1.658373355563035, −0.6306544289914960, 0,
0.6306544289914960, 1.658373355563035, 1.981808336726995, 2.564438762659587, 3.421071910556062, 3.676993012198466, 4.211787086444181, 4.906483929190479, 5.161919225431572, 5.669875818870397, 6.179104781785633, 6.786881689512721, 7.018075787086882, 7.404813807275197, 8.051497099318524, 8.652404085536158, 9.315652404807452, 9.443878108926148, 10.09725067445848, 10.59948992095448, 11.02022764563857, 11.34791874277799, 11.95929998491538, 12.34454737519852, 12.51466845529548