| L(s) = 1 | + 1.91e3·2-s − 5.90e4·3-s + 1.55e6·4-s − 1.12e8·6-s + 1.48e8·7-s − 1.03e9·8-s + 3.48e9·9-s + 1.36e11·11-s − 9.17e10·12-s + 5.60e11·13-s + 2.83e11·14-s − 5.24e12·16-s − 1.60e13·17-s + 6.66e12·18-s + 4.50e13·19-s − 8.77e12·21-s + 2.60e14·22-s − 1.52e14·23-s + 6.13e13·24-s + 1.07e15·26-s − 2.05e14·27-s + 2.30e14·28-s + 2.68e15·29-s − 2.10e15·31-s − 7.83e15·32-s − 8.05e15·33-s − 3.06e16·34-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 0.577·3-s + 0.740·4-s − 0.761·6-s + 0.198·7-s − 0.342·8-s + 0.333·9-s + 1.58·11-s − 0.427·12-s + 1.12·13-s + 0.262·14-s − 1.19·16-s − 1.92·17-s + 0.439·18-s + 1.68·19-s − 0.114·21-s + 2.09·22-s − 0.765·23-s + 0.197·24-s + 1.48·26-s − 0.192·27-s + 0.147·28-s + 1.18·29-s − 0.460·31-s − 1.23·32-s − 0.915·33-s − 2.54·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(4.264619870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.264619870\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.91e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 1.48e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.36e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 5.60e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.60e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.50e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.52e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.68e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.10e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.10e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.74e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.16e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.56e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.00e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 3.90e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 9.66e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.77e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 5.04e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.34e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 7.34e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.78e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.57e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.34e20T + 5.27e41T^{2} \) |
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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
−11.37940693748591910982853255936, −9.613883481327292047964137937195, −8.555434625770197511591687451456, −6.73968824897499610480769981278, −6.25432169813242528024392965635, −5.07457347728529220995262200867, −4.17440426235699619926642773158, −3.39632857401460406604528075568, −1.88428314701229523235116282577, −0.73854942913207828854717027817,
0.73854942913207828854717027817, 1.88428314701229523235116282577, 3.39632857401460406604528075568, 4.17440426235699619926642773158, 5.07457347728529220995262200867, 6.25432169813242528024392965635, 6.73968824897499610480769981278, 8.555434625770197511591687451456, 9.613883481327292047964137937195, 11.37940693748591910982853255936
