| L(s) = 1 | + 4·2-s + 19.3·3-s + 16·4-s + 25·5-s + 77.2·6-s + 49·7-s + 64·8-s + 129.·9-s + 100·10-s − 10.9·11-s + 308.·12-s + 29.6·13-s + 196·14-s + 482.·15-s + 256·16-s − 432.·17-s + 518.·18-s − 956.·19-s + 400·20-s + 945.·21-s − 43.6·22-s + 979.·23-s + 1.23e3·24-s + 625·25-s + 118.·26-s − 2.19e3·27-s + 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.23·3-s + 0.5·4-s + 0.447·5-s + 0.875·6-s + 0.377·7-s + 0.353·8-s + 0.532·9-s + 0.316·10-s − 0.0271·11-s + 0.619·12-s + 0.0487·13-s + 0.267·14-s + 0.553·15-s + 0.250·16-s − 0.362·17-s + 0.376·18-s − 0.607·19-s + 0.223·20-s + 0.467·21-s − 0.0192·22-s + 0.386·23-s + 0.437·24-s + 0.200·25-s + 0.0344·26-s − 0.578·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.106971135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.106971135\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 19.3T + 243T^{2} \) |
| 11 | \( 1 + 10.9T + 1.61e5T^{2} \) |
| 13 | \( 1 - 29.6T + 3.71e5T^{2} \) |
| 17 | \( 1 + 432.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 956.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 979.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 996.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 287.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.34e4T + 8.58e9T^{2} \) |
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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
−13.74841480714087636429073542884, −13.06249794604849262778061579849, −11.65604083656017771370309373845, −10.30336074311726783145385429900, −8.967476484055572232393607599248, −7.933494412902646523420663161919, −6.45089084514856190986276310439, −4.79425281178258313061634131097, −3.24421292151904464068621847891, −1.94793016459663574604823683788,
1.94793016459663574604823683788, 3.24421292151904464068621847891, 4.79425281178258313061634131097, 6.45089084514856190986276310439, 7.933494412902646523420663161919, 8.967476484055572232393607599248, 10.30336074311726783145385429900, 11.65604083656017771370309373845, 13.06249794604849262778061579849, 13.74841480714087636429073542884
