| L(s) = 1 | − 8·2-s + 61.8·3-s + 64·4-s − 495.·6-s + 343·7-s − 512·8-s + 1.64e3·9-s + 4.83e3·11-s + 3.96e3·12-s + 1.26e4·13-s − 2.74e3·14-s + 4.09e3·16-s + 8.84e3·17-s − 1.31e4·18-s + 4.84e4·19-s + 2.12e4·21-s − 3.86e4·22-s − 7.59e3·23-s − 3.16e4·24-s − 1.00e5·26-s − 3.36e4·27-s + 2.19e4·28-s + 5.17e4·29-s − 2.11e5·31-s − 3.27e4·32-s + 2.99e5·33-s − 7.07e4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.935·6-s + 0.377·7-s − 0.353·8-s + 0.751·9-s + 1.09·11-s + 0.661·12-s + 1.59·13-s − 0.267·14-s + 0.250·16-s + 0.436·17-s − 0.531·18-s + 1.62·19-s + 0.500·21-s − 0.774·22-s − 0.130·23-s − 0.467·24-s − 1.12·26-s − 0.328·27-s + 0.188·28-s + 0.394·29-s − 1.27·31-s − 0.176·32-s + 1.44·33-s − 0.308·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.554077659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.554077659\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 61.8T + 2.18e3T^{2} \) |
| 11 | \( 1 - 4.83e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.26e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 8.84e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.84e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.59e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.17e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.94e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.30e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.36e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.70e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.92e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.01e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.92e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.36e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.85e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.24e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.22e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.04e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.18e6T + 8.07e13T^{2} \) |
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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
−9.943306702379864646823160199770, −9.165163771755191103754785513500, −8.540569700420179066008771270255, −7.79124779002345368391958655927, −6.80302027412133891411246262835, −5.57947375939889018895749579362, −3.83855322392834426757354082167, −3.18955872991859768875449016744, −1.78782307999975967131364370630, −1.02722161905687690798759284066,
1.02722161905687690798759284066, 1.78782307999975967131364370630, 3.18955872991859768875449016744, 3.83855322392834426757354082167, 5.57947375939889018895749579362, 6.80302027412133891411246262835, 7.79124779002345368391958655927, 8.540569700420179066008771270255, 9.165163771755191103754785513500, 9.943306702379864646823160199770
