| L(s) = 1 | − 0.806·2-s + 22.8·3-s − 31.3·4-s + 25·5-s − 18.4·6-s − 91.7·7-s + 51.0·8-s + 281.·9-s − 20.1·10-s − 121·11-s − 717.·12-s − 55.9·13-s + 73.9·14-s + 572.·15-s + 961.·16-s − 1.87e3·17-s − 226.·18-s − 361·19-s − 783.·20-s − 2.10e3·21-s + 97.5·22-s − 3.67e3·23-s + 1.16e3·24-s + 625·25-s + 45.1·26-s + 878.·27-s + 2.87e3·28-s + ⋯ |
| L(s) = 1 | − 0.142·2-s + 1.46·3-s − 0.979·4-s + 0.447·5-s − 0.209·6-s − 0.707·7-s + 0.282·8-s + 1.15·9-s − 0.0637·10-s − 0.301·11-s − 1.43·12-s − 0.0918·13-s + 0.100·14-s + 0.656·15-s + 0.939·16-s − 1.57·17-s − 0.165·18-s − 0.229·19-s − 0.438·20-s − 1.03·21-s + 0.0429·22-s − 1.44·23-s + 0.414·24-s + 0.200·25-s + 0.0130·26-s + 0.231·27-s + 0.693·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.238991265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238991265\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 0.806T + 32T^{2} \) |
| 3 | \( 1 - 22.8T + 243T^{2} \) |
| 7 | \( 1 + 91.7T + 1.68e4T^{2} \) |
| 13 | \( 1 + 55.9T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.87e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.58e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.93e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.96e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.09e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.06e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.37e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.17e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.26e4T + 8.58e9T^{2} \) |
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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
−9.095309882480642060191126387802, −8.493968655974191089918066310776, −7.911284389925722810713817043593, −6.78011350970590609358339443365, −5.82493676891940974160592614448, −4.51461391613651526342541428907, −3.90472194320984973036052094834, −2.78462845702615420619866185189, −2.07995542948679057062331400080, −0.59690484692879169408406657999,
0.59690484692879169408406657999, 2.07995542948679057062331400080, 2.78462845702615420619866185189, 3.90472194320984973036052094834, 4.51461391613651526342541428907, 5.82493676891940974160592614448, 6.78011350970590609358339443365, 7.911284389925722810713817043593, 8.493968655974191089918066310776, 9.095309882480642060191126387802
