| 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 − 3·17-s + 2·18-s − 5·19-s + 20-s − 21-s + 2·22-s + 4·23-s + 24-s + 25-s + 5·26-s + 5·27-s + 28-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 − 0.727·17-s + 0.471·18-s − 1.14·19-s + 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s + 0.962·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−15.24514138251980, −14.93872722012921, −14.34629210747548, −13.73566262946394, −13.19184225633638, −12.50101529000292, −12.13474852888660, −11.55689850169186, −10.98072361624459, −10.57299564139429, −10.14939297342147, −9.537144575277013, −8.763563356101776, −8.590954456612275, −7.862612099046117, −7.209328401120816, −6.635100199505160, −6.269041730255042, −5.278081431953229, −5.124000656759516, −4.478672559876515, −3.355461428235752, −2.664088493071863, −2.127254126686609, −1.397463004086718, 0, 0,
1.397463004086718, 2.127254126686609, 2.664088493071863, 3.355461428235752, 4.478672559876515, 5.124000656759516, 5.278081431953229, 6.269041730255042, 6.635100199505160, 7.209328401120816, 7.862612099046117, 8.590954456612275, 8.763563356101776, 9.537144575277013, 10.14939297342147, 10.57299564139429, 10.98072361624459, 11.55689850169186, 12.13474852888660, 12.50101529000292, 13.19184225633638, 13.73566262946394, 14.34629210747548, 14.93872722012921, 15.24514138251980
