| L(s) = 1 | − 169.·2-s − 1.02e5·4-s − 2.91e5·5-s − 5.76e6·7-s + 3.95e7·8-s + 4.95e7·10-s − 9.20e8·11-s − 3.54e9·13-s + 9.78e8·14-s + 6.68e9·16-s − 1.33e10·17-s + 1.01e11·19-s + 2.98e10·20-s + 1.56e11·22-s − 5.71e11·23-s − 6.77e11·25-s + 6.02e11·26-s + 5.89e11·28-s − 2.09e12·29-s − 7.36e12·31-s − 6.32e12·32-s + 2.26e12·34-s + 1.68e12·35-s + 4.05e13·37-s − 1.72e13·38-s − 1.15e13·40-s − 6.41e13·41-s + ⋯ |
| L(s) = 1 | − 0.468·2-s − 0.780·4-s − 0.334·5-s − 0.377·7-s + 0.834·8-s + 0.156·10-s − 1.29·11-s − 1.20·13-s + 0.177·14-s + 0.389·16-s − 0.464·17-s + 1.37·19-s + 0.260·20-s + 0.606·22-s − 1.52·23-s − 0.888·25-s + 0.565·26-s + 0.294·28-s − 0.776·29-s − 1.55·31-s − 1.01·32-s + 0.217·34-s + 0.126·35-s + 1.90·37-s − 0.642·38-s − 0.278·40-s − 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.01407042501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01407042501\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 5.76e6T \) |
| good | 2 | \( 1 + 169.T + 1.31e5T^{2} \) |
| 5 | \( 1 + 2.91e5T + 7.62e11T^{2} \) |
| 11 | \( 1 + 9.20e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.54e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 1.33e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.01e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 5.71e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.09e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 7.36e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 4.05e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.41e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 6.38e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 2.90e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 5.20e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 8.11e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 7.20e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.00e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 1.62e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.98e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.79e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.19e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.90e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.12e17T + 5.95e33T^{2} \) |
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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
−11.47505510246946902940627622982, −10.05292550111507919675121697139, −9.529444804018039992216487963654, −8.047433044486219366557425980455, −7.44446685573931629560840508174, −5.60948229797532948325149734463, −4.60462121461916184940791811503, −3.30003528962594154023633718253, −1.85876936668587339952493689592, −0.05765367944186395641909138104,
0.05765367944186395641909138104, 1.85876936668587339952493689592, 3.30003528962594154023633718253, 4.60462121461916184940791811503, 5.60948229797532948325149734463, 7.44446685573931629560840508174, 8.047433044486219366557425980455, 9.529444804018039992216487963654, 10.05292550111507919675121697139, 11.47505510246946902940627622982
