| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 25·5-s + 36·6-s + 9.46·7-s + 64·8-s + 81·9-s − 100·10-s + 715.·11-s + 144·12-s + 81.7·13-s + 37.8·14-s − 225·15-s + 256·16-s − 218.·17-s + 324·18-s + 361·19-s − 400·20-s + 85.2·21-s + 2.86e3·22-s + 879.·23-s + 576·24-s + 625·25-s + 327.·26-s + 729·27-s + 151.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.0730·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.78·11-s + 0.288·12-s + 0.134·13-s + 0.0516·14-s − 0.258·15-s + 0.250·16-s − 0.183·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.0421·21-s + 1.26·22-s + 0.346·23-s + 0.204·24-s + 0.200·25-s + 0.0949·26-s + 0.192·27-s + 0.0365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.848905057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.848905057\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 - 9.46T + 1.68e4T^{2} \) |
| 11 | \( 1 - 715.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 81.7T + 3.71e5T^{2} \) |
| 17 | \( 1 + 218.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 879.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.73e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 487.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.92e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.42e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.59e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.96e4T + 8.58e9T^{2} \) |
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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
−9.836931015488215994597764604340, −9.069792953196941553205723008014, −8.154686942973809727545564023107, −7.11311776767456755808633803011, −6.44337800703815285649470795761, −5.19511252885928129468831741144, −4.01213578923593077871719744370, −3.54863273085969843524228544692, −2.16140431004886646602906679473, −1.00179684755921844291343451786,
1.00179684755921844291343451786, 2.16140431004886646602906679473, 3.54863273085969843524228544692, 4.01213578923593077871719744370, 5.19511252885928129468831741144, 6.44337800703815285649470795761, 7.11311776767456755808633803011, 8.154686942973809727545564023107, 9.069792953196941553205723008014, 9.836931015488215994597764604340
