| L(s) = 1 | − 131.·2-s − 395.·3-s + 9.05e3·4-s + 3.44e4·5-s + 5.18e4·6-s + 3.74e5·7-s − 1.13e5·8-s − 1.43e6·9-s − 4.52e6·10-s + 6.52e6·11-s − 3.57e6·12-s − 4.92e7·14-s − 1.36e7·15-s − 5.92e7·16-s − 1.13e8·17-s + 1.88e8·18-s − 7.26e7·19-s + 3.11e8·20-s − 1.48e8·21-s − 8.56e8·22-s − 2.61e8·23-s + 4.47e7·24-s − 3.50e7·25-s + 1.19e9·27-s + 3.39e9·28-s − 4.36e9·29-s + 1.78e9·30-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.312·3-s + 1.10·4-s + 0.985·5-s + 0.454·6-s + 1.20·7-s − 0.152·8-s − 0.902·9-s − 1.43·10-s + 1.11·11-s − 0.345·12-s − 1.74·14-s − 0.308·15-s − 0.883·16-s − 1.13·17-s + 1.30·18-s − 0.354·19-s + 1.08·20-s − 0.376·21-s − 1.61·22-s − 0.368·23-s + 0.0478·24-s − 0.0286·25-s + 0.595·27-s + 1.33·28-s − 1.36·29-s + 0.447·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.084274288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084274288\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 131.T + 8.19e3T^{2} \) |
| 3 | \( 1 + 395.T + 1.59e6T^{2} \) |
| 5 | \( 1 - 3.44e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 3.74e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 6.52e6T + 3.45e13T^{2} \) |
| 17 | \( 1 + 1.13e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 7.26e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 2.61e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.36e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 2.46e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.21e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.26e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.89e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.30e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 9.35e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 3.85e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 1.30e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.00e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.79e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.44e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.00e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.32e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 9.30e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 9.20e12T + 6.73e25T^{2} \) |
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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
−10.33489479574596603997136240500, −9.140153225920176225124149954472, −8.775723074623973310349022379971, −7.64048230153423503764198383915, −6.49743982242921808547447400804, −5.51944831841184912465136539668, −4.23438064606849692429910680613, −2.22805537141379520003796694391, −1.67804326106117589659761135441, −0.57507440345579771256979244586,
0.57507440345579771256979244586, 1.67804326106117589659761135441, 2.22805537141379520003796694391, 4.23438064606849692429910680613, 5.51944831841184912465136539668, 6.49743982242921808547447400804, 7.64048230153423503764198383915, 8.775723074623973310349022379971, 9.140153225920176225124149954472, 10.33489479574596603997136240500
