L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 5·7-s + 8-s − 2·9-s − 10-s − 6·11-s + 12-s + 5·13-s + 5·14-s − 15-s + 16-s − 3·17-s − 2·18-s − 19-s − 20-s + 5·21-s − 6·22-s + 6·23-s + 24-s + 25-s + 5·26-s − 5·27-s + 5·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s + 1.38·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.223·20-s + 1.09·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.962·27-s + 0.944·28-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
\(L(1)\) |
\(\approx\) |
\(5.049961571\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.049961571\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.96372785152562, −14.15896832336359, −13.74364637313069, −13.40778787321319, −12.83865743970018, −12.24009862713436, −11.40310304950896, −11.18244649505932, −10.83530984840432, −10.41781207254362, −9.253420073756281, −8.662662429538813, −8.342799931307725, −7.829896050258661, −7.467080693708797, −6.684419965805504, −5.721528653069111, −5.471708472739204, −4.763687899415500, −4.348368179549606, −3.575190122424362, −2.870613257891162, −2.341074911512905, −1.667505494455712, −0.6941483256393399,
0.6941483256393399, 1.667505494455712, 2.341074911512905, 2.870613257891162, 3.575190122424362, 4.348368179549606, 4.763687899415500, 5.471708472739204, 5.721528653069111, 6.684419965805504, 7.467080693708797, 7.829896050258661, 8.342799931307725, 8.662662429538813, 9.253420073756281, 10.41781207254362, 10.83530984840432, 11.18244649505932, 11.40310304950896, 12.24009862713436, 12.83865743970018, 13.40778787321319, 13.74364637313069, 14.15896832336359, 14.96372785152562