| L(s) = 1 | − 51.2·2-s + 1.73e3·3-s − 1.28e5·4-s + 1.90e5·5-s − 8.91e4·6-s + 5.76e6·7-s + 1.33e7·8-s − 1.26e8·9-s − 9.78e6·10-s + 8.49e8·11-s − 2.23e8·12-s + 3.73e9·13-s − 2.95e8·14-s + 3.31e8·15-s + 1.61e10·16-s + 3.16e10·17-s + 6.46e9·18-s + 5.89e10·19-s − 2.44e10·20-s + 1.00e10·21-s − 4.35e10·22-s + 1.00e11·23-s + 2.31e10·24-s − 7.26e11·25-s − 1.91e11·26-s − 4.43e11·27-s − 7.40e11·28-s + ⋯ |
| L(s) = 1 | − 0.141·2-s + 0.152·3-s − 0.979·4-s + 0.218·5-s − 0.0216·6-s + 0.377·7-s + 0.280·8-s − 0.976·9-s − 0.0309·10-s + 1.19·11-s − 0.149·12-s + 1.26·13-s − 0.0535·14-s + 0.0334·15-s + 0.940·16-s + 1.09·17-s + 0.138·18-s + 0.795·19-s − 0.213·20-s + 0.0578·21-s − 0.169·22-s + 0.268·23-s + 0.0429·24-s − 0.952·25-s − 0.179·26-s − 0.302·27-s − 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.586174443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586174443\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 5.76e6T \) |
| good | 2 | \( 1 + 51.2T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.73e3T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.90e5T + 7.62e11T^{2} \) |
| 11 | \( 1 - 8.49e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.73e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.16e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 5.89e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.00e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 4.00e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 6.91e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.60e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 2.32e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 2.01e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.16e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.19e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.86e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 8.63e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 3.23e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 4.28e14T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.32e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 8.04e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.14e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 5.68e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.50e17T + 5.95e33T^{2} \) |
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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
−17.93655364231331515730214565564, −16.79529551831800228308917008048, −14.56572067607887765948765286439, −13.58077149296326531181788547533, −11.57704273961363925959385789663, −9.518599898445053886657675220400, −8.236572155779179862891645654029, −5.69919697816253015251490747652, −3.69840921948115600336909107279, −1.07649852127327007727264827145,
1.07649852127327007727264827145, 3.69840921948115600336909107279, 5.69919697816253015251490747652, 8.236572155779179862891645654029, 9.518599898445053886657675220400, 11.57704273961363925959385789663, 13.58077149296326531181788547533, 14.56572067607887765948765286439, 16.79529551831800228308917008048, 17.93655364231331515730214565564
