| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 36·6-s − 233.·7-s − 64·8-s + 81·9-s − 89.7·11-s − 144·12-s + 209.·13-s + 934.·14-s + 256·16-s − 226.·17-s − 324·18-s − 2.18e3·19-s + 2.10e3·21-s + 358.·22-s − 4.29e3·23-s + 576·24-s − 836.·26-s − 729·27-s − 3.73e3·28-s + 3.55e3·29-s + 6.90e3·31-s − 1.02e3·32-s + 807.·33-s + 907.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.80·7-s − 0.353·8-s + 0.333·9-s − 0.223·11-s − 0.288·12-s + 0.343·13-s + 1.27·14-s + 0.250·16-s − 0.190·17-s − 0.235·18-s − 1.38·19-s + 1.04·21-s + 0.158·22-s − 1.69·23-s + 0.204·24-s − 0.242·26-s − 0.192·27-s − 0.901·28-s + 0.785·29-s + 1.28·31-s − 0.176·32-s + 0.129·33-s + 0.134·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5566613547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5566613547\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 233.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 89.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 209.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 226.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.71e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.96e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.13e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.84e4T + 8.58e9T^{2} \) |
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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
−12.17436017767392372868996157004, −10.81367759770916395760676759348, −10.08159364564766380779443747136, −9.209030885003342525843328707388, −7.939026226457516721016722079514, −6.49980192183176676728689764585, −6.09389312706301102454665403602, −4.08163216225521999895733547917, −2.53407190524941987297917211445, −0.53280970123020234144907439063,
0.53280970123020234144907439063, 2.53407190524941987297917211445, 4.08163216225521999895733547917, 6.09389312706301102454665403602, 6.49980192183176676728689764585, 7.939026226457516721016722079514, 9.209030885003342525843328707388, 10.08159364564766380779443747136, 10.81367759770916395760676759348, 12.17436017767392372868996157004
