| L(s) = 1 | + 4·2-s + 16.6·3-s + 16·4-s + 46.0·5-s + 66.5·6-s + 16.5·7-s + 64·8-s + 34.1·9-s + 184.·10-s + 514.·11-s + 266.·12-s − 1.10e3·13-s + 66.1·14-s + 766.·15-s + 256·16-s + 1.15e3·17-s + 136.·18-s − 1.49e3·19-s + 736.·20-s + 275.·21-s + 2.05e3·22-s + 3.06e3·23-s + 1.06e3·24-s − 1.00e3·25-s − 4.40e3·26-s − 3.47e3·27-s + 264.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.06·3-s + 0.5·4-s + 0.823·5-s + 0.755·6-s + 0.127·7-s + 0.353·8-s + 0.140·9-s + 0.582·10-s + 1.28·11-s + 0.533·12-s − 1.80·13-s + 0.0902·14-s + 0.879·15-s + 0.250·16-s + 0.966·17-s + 0.0993·18-s − 0.952·19-s + 0.411·20-s + 0.136·21-s + 0.906·22-s + 1.20·23-s + 0.377·24-s − 0.321·25-s − 1.27·26-s − 0.917·27-s + 0.0638·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.116133820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.116133820\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 37 | \( 1 + 1.36e3T \) |
| good | 3 | \( 1 - 16.6T + 243T^{2} \) |
| 5 | \( 1 - 46.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 16.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 514.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.10e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.15e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.32e3T + 2.86e7T^{2} \) |
| 41 | \( 1 - 9.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 172.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.97e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.09e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.69e4T + 8.58e9T^{2} \) |
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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
−13.84517823030856306177284876530, −12.72805399988035368223371348902, −11.65437405509230339152595863093, −9.998392555773937025026491078001, −9.132069830562385137968453112995, −7.69359351906615319697410666501, −6.34125199032074962812099885236, −4.80983134210244023011291018253, −3.17296304773183854232197323536, −1.90590123025953163547925026449,
1.90590123025953163547925026449, 3.17296304773183854232197323536, 4.80983134210244023011291018253, 6.34125199032074962812099885236, 7.69359351906615319697410666501, 9.132069830562385137968453112995, 9.998392555773937025026491078001, 11.65437405509230339152595863093, 12.72805399988035368223371348902, 13.84517823030856306177284876530
