L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 4·7-s + 8-s + 9-s − 10-s − 12-s − 13-s + 4·14-s + 15-s + 16-s + 2·17-s + 18-s − 4·19-s − 20-s − 4·21-s + 8·23-s − 24-s + 25-s − 26-s − 27-s + 4·28-s + 30-s + 8·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s + 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.868009425\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.868009425\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.64603679781798, −11.95106006586379, −11.69026094466167, −11.49572869682921, −10.85013863614599, −10.51002633745729, −10.18164193337369, −9.429957683932685, −8.761115990070720, −8.400521103756670, −7.958634480432956, −7.404633631033155, −7.030085396305424, −6.516577466122280, −5.999207819556739, −5.300605484897704, −5.030670342375546, −4.653269741213946, −4.172480468160605, −3.632281347912578, −2.910582897355153, −2.400708971525416, −1.713541887752158, −1.146580219894488, −0.5743976173162840,
0.5743976173162840, 1.146580219894488, 1.713541887752158, 2.400708971525416, 2.910582897355153, 3.632281347912578, 4.172480468160605, 4.653269741213946, 5.030670342375546, 5.300605484897704, 5.999207819556739, 6.516577466122280, 7.030085396305424, 7.404633631033155, 7.958634480432956, 8.400521103756670, 8.761115990070720, 9.429957683932685, 10.18164193337369, 10.51002633745729, 10.85013863614599, 11.49572869682921, 11.69026094466167, 11.95106006586379, 12.64603679781798