| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 4.85·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s + 9.70·10-s − 37.3·11-s + 12·12-s − 13·13-s − 14·14-s − 14.5·15-s + 16·16-s + 117.·17-s − 18·18-s − 94.5·19-s − 19.4·20-s + 21·21-s + 74.6·22-s + 217.·23-s − 24·24-s − 101.·25-s + 26·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.434·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.307·10-s − 1.02·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.250·15-s + 0.250·16-s + 1.67·17-s − 0.235·18-s − 1.14·19-s − 0.217·20-s + 0.218·21-s + 0.723·22-s + 1.97·23-s − 0.204·24-s − 0.811·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.570883320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570883320\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 4.85T + 125T^{2} \) |
| 11 | \( 1 + 37.3T + 1.33e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 217.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 82.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 19.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 405.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 12.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 293.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 331.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 703.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 676.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 541.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 516.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.58e3T + 9.12e5T^{2} \) |
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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
−10.43611511553713195708241795218, −9.402680116719614762278544562674, −8.570469629063550638751874169826, −7.74407895757037090225701991948, −7.27654741611401354425173202815, −5.83995636640060591330507407821, −4.70538491132729737906645356052, −3.34229731717722613794635496913, −2.31623863882034444026476157599, −0.829724411253698698526325957471,
0.829724411253698698526325957471, 2.31623863882034444026476157599, 3.34229731717722613794635496913, 4.70538491132729737906645356052, 5.83995636640060591330507407821, 7.27654741611401354425173202815, 7.74407895757037090225701991948, 8.570469629063550638751874169826, 9.402680116719614762278544562674, 10.43611511553713195708241795218
