| L(s) = 1 | + 4.52·2-s + 12.4·4-s + 30.8·7-s + 20.3·8-s − 6.79·11-s + 31.2·13-s + 139.·14-s − 7.92·16-s + 112.·17-s − 60.9·19-s − 30.7·22-s − 31.1·23-s + 141.·26-s + 384.·28-s + 189.·29-s − 343.·31-s − 198.·32-s + 508.·34-s + 206.·37-s − 275.·38-s + 435.·41-s + 60.9·43-s − 84.8·44-s − 141.·46-s + 251.·47-s + 606.·49-s + 390.·52-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.56·4-s + 1.66·7-s + 0.898·8-s − 0.186·11-s + 0.666·13-s + 2.66·14-s − 0.123·16-s + 1.60·17-s − 0.735·19-s − 0.298·22-s − 0.282·23-s + 1.06·26-s + 2.59·28-s + 1.21·29-s − 1.99·31-s − 1.09·32-s + 2.56·34-s + 0.917·37-s − 1.17·38-s + 1.65·41-s + 0.216·43-s − 0.290·44-s − 0.452·46-s + 0.779·47-s + 1.76·49-s + 1.04·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.376966069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.376966069\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4.52T + 8T^{2} \) |
| 7 | \( 1 - 30.8T + 343T^{2} \) |
| 11 | \( 1 + 6.79T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 60.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 31.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 343.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 435.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 60.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 248.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 571.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 329.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 677.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 238.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 826.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.30e3T + 9.12e5T^{2} \) |
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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
−10.67316399613602434547785804389, −9.158464064031101008964837610454, −8.076004724657294320409438458777, −7.40499819074119689198478181986, −6.06550142616669725385916639829, −5.45421313742231050988344290478, −4.56275517336600014147039982849, −3.79646668460428703483004766124, −2.53852724775268535770394899417, −1.34709346063123769519462392363,
1.34709346063123769519462392363, 2.53852724775268535770394899417, 3.79646668460428703483004766124, 4.56275517336600014147039982849, 5.45421313742231050988344290478, 6.06550142616669725385916639829, 7.40499819074119689198478181986, 8.076004724657294320409438458777, 9.158464064031101008964837610454, 10.67316399613602434547785804389
