| L(s) = 1 | + 39.0·2-s − 174.·3-s + 1.01e3·4-s − 303.·5-s − 6.80e3·6-s + 1.25e4·7-s + 1.95e4·8-s + 1.07e4·9-s − 1.18e4·10-s + 2.80e4·11-s − 1.76e5·12-s + 8.30e4·13-s + 4.90e5·14-s + 5.29e4·15-s + 2.45e5·16-s + 5.97e5·17-s + 4.18e5·18-s − 1.24e5·19-s − 3.07e5·20-s − 2.18e6·21-s + 1.09e6·22-s − 1.57e6·23-s − 3.41e6·24-s − 1.86e6·25-s + 3.24e6·26-s + 1.56e6·27-s + 1.27e7·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 1.24·3-s + 1.97·4-s − 0.217·5-s − 2.14·6-s + 1.97·7-s + 1.68·8-s + 0.543·9-s − 0.375·10-s + 0.577·11-s − 2.45·12-s + 0.806·13-s + 3.41·14-s + 0.270·15-s + 0.936·16-s + 1.73·17-s + 0.938·18-s − 0.218·19-s − 0.430·20-s − 2.45·21-s + 0.997·22-s − 1.17·23-s − 2.09·24-s − 0.952·25-s + 1.39·26-s + 0.566·27-s + 3.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.993880380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.993880380\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 7.07e5T \) |
| good | 2 | \( 1 - 39.0T + 512T^{2} \) |
| 3 | \( 1 + 174.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 303.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.25e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.80e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 8.30e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.97e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.24e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.57e6T + 1.80e12T^{2} \) |
| 31 | \( 1 + 4.96e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.43e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.97e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.23e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.50e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.75e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.26e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.95e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.79e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.11e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.73e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.29e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.59e8T + 7.60e17T^{2} \) |
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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
−14.67198516911082760059301836117, −14.00433014870671600291953267722, −12.24149495374628533207996319426, −11.63826148615711051340406794397, −10.84272067829767106790706394316, −7.83319491723503748326156433596, −6.04392478483799008576153510785, −5.18384266044858084189341895291, −3.98536306303619386823235668624, −1.51281490390107966481586227337,
1.51281490390107966481586227337, 3.98536306303619386823235668624, 5.18384266044858084189341895291, 6.04392478483799008576153510785, 7.83319491723503748326156433596, 10.84272067829767106790706394316, 11.63826148615711051340406794397, 12.24149495374628533207996319426, 14.00433014870671600291953267722, 14.67198516911082760059301836117
