| L(s) = 1 | + 4·2-s + 18.3·3-s + 16·4-s + 42.3·5-s + 73.5·6-s − 127.·7-s + 64·8-s + 95.2·9-s + 169.·10-s + 381.·11-s + 294.·12-s − 726.·13-s − 508.·14-s + 778.·15-s + 256·16-s + 1.05e3·17-s + 381.·18-s − 361·19-s + 677.·20-s − 2.34e3·21-s + 1.52e3·22-s − 1.56e3·23-s + 1.17e3·24-s − 1.33e3·25-s − 2.90e3·26-s − 2.71e3·27-s − 2.03e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.17·3-s + 0.5·4-s + 0.757·5-s + 0.834·6-s − 0.981·7-s + 0.353·8-s + 0.392·9-s + 0.535·10-s + 0.951·11-s + 0.589·12-s − 1.19·13-s − 0.693·14-s + 0.893·15-s + 0.250·16-s + 0.888·17-s + 0.277·18-s − 0.229·19-s + 0.378·20-s − 1.15·21-s + 0.672·22-s − 0.617·23-s + 0.417·24-s − 0.426·25-s − 0.843·26-s − 0.717·27-s − 0.490·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.275038737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.275038737\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 - 18.3T + 243T^{2} \) |
| 5 | \( 1 - 42.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 127.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 381.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 726.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.05e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 740.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 457.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.25e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.19e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.79e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.77e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.73e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.79e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.57e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.67e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.63e5T + 8.58e9T^{2} \) |
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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
−14.77001991605762824003177925530, −14.21471844646158847241960606128, −13.16610802949862589190098847049, −12.06871797728980103274202703439, −10.03083258933875487474879828714, −9.141146289656077346440226666474, −7.35902114747356138444374497877, −5.84528092309695604651295984623, −3.70490709866047953101249732251, −2.29921338885678843843024970314,
2.29921338885678843843024970314, 3.70490709866047953101249732251, 5.84528092309695604651295984623, 7.35902114747356138444374497877, 9.141146289656077346440226666474, 10.03083258933875487474879828714, 12.06871797728980103274202703439, 13.16610802949862589190098847049, 14.21471844646158847241960606128, 14.77001991605762824003177925530
