| L(s) = 1 | + 21.3·2-s − 81·3-s − 55.0·4-s − 625·5-s − 1.73e3·6-s + 6.10e3·7-s − 1.21e4·8-s + 6.56e3·9-s − 1.33e4·10-s + 622.·11-s + 4.46e3·12-s − 1.44e4·13-s + 1.30e5·14-s + 5.06e4·15-s − 2.30e5·16-s + 4.38e5·17-s + 1.40e5·18-s + 1.30e5·19-s + 3.44e4·20-s − 4.94e5·21-s + 1.33e4·22-s − 2.57e6·23-s + 9.81e5·24-s + 3.90e5·25-s − 3.09e5·26-s − 5.31e5·27-s − 3.36e5·28-s + ⋯ |
| L(s) = 1 | + 0.944·2-s − 0.577·3-s − 0.107·4-s − 0.447·5-s − 0.545·6-s + 0.961·7-s − 1.04·8-s + 0.333·9-s − 0.422·10-s + 0.0128·11-s + 0.0621·12-s − 0.140·13-s + 0.908·14-s + 0.258·15-s − 0.880·16-s + 1.27·17-s + 0.314·18-s + 0.229·19-s + 0.0481·20-s − 0.555·21-s + 0.0121·22-s − 1.92·23-s + 0.604·24-s + 0.200·25-s − 0.132·26-s − 0.192·27-s − 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.011896691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011896691\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 + 625T \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 2 | \( 1 - 21.3T + 512T^{2} \) |
| 7 | \( 1 - 6.10e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 622.T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.44e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.38e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 2.57e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.25e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.41e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.41e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.50e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.79e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.12e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.94e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.60e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.21e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.25e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.78e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.71e9T + 7.60e17T^{2} \) |
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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
−10.46535764611806231235892369586, −9.352986371068976090734051966544, −8.170423506300906151350295268808, −7.31538026730037089614347020057, −5.85707384360691699275139992010, −5.31187303398777377552895854194, −4.26839401778169326461589800059, −3.51912994798924211648494547970, −1.93919887950630006436407520917, −0.56240883126517642510858289053,
0.56240883126517642510858289053, 1.93919887950630006436407520917, 3.51912994798924211648494547970, 4.26839401778169326461589800059, 5.31187303398777377552895854194, 5.85707384360691699275139992010, 7.31538026730037089614347020057, 8.170423506300906151350295268808, 9.352986371068976090734051966544, 10.46535764611806231235892369586
