| L(s) = 1 | − 17.8·2-s − 27·3-s + 188.·4-s − 168.·5-s + 480.·6-s − 317.·7-s − 1.08e3·8-s + 729·9-s + 2.99e3·10-s − 3.14e3·11-s − 5.09e3·12-s + 1.27e4·13-s + 5.64e3·14-s + 4.54e3·15-s − 4.88e3·16-s + 1.85e4·17-s − 1.29e4·18-s + 1.91e4·19-s − 3.17e4·20-s + 8.56e3·21-s + 5.60e4·22-s + 5.56e4·23-s + 2.92e4·24-s − 4.98e4·25-s − 2.26e5·26-s − 1.96e4·27-s − 5.99e4·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s − 0.577·3-s + 1.47·4-s − 0.601·5-s + 0.908·6-s − 0.349·7-s − 0.748·8-s + 0.333·9-s + 0.946·10-s − 0.713·11-s − 0.851·12-s + 1.60·13-s + 0.550·14-s + 0.347·15-s − 0.298·16-s + 0.916·17-s − 0.524·18-s + 0.640·19-s − 0.888·20-s + 0.201·21-s + 1.12·22-s + 0.954·23-s + 0.432·24-s − 0.637·25-s − 2.52·26-s − 0.192·27-s − 0.515·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7922507051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7922507051\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 + 17.8T + 128T^{2} \) |
| 5 | \( 1 + 168.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 317.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.14e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.27e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.85e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.91e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.56e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.88e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.73e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.82e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.45e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.30e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.76e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.03e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.08e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.73e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.60e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.38e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.07e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.00e6T + 8.07e13T^{2} \) |
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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
−9.895668854225745190025226420581, −8.941572012338263859574986210825, −8.118222509118806205369521353713, −7.42840002210079633494617822779, −6.46728649234214096330961986722, −5.44558096460866677181328202197, −4.00461985545960213412474245634, −2.75602850459425458663151904508, −1.16566286808580849780323258043, −0.64198825919724980231029857034,
0.64198825919724980231029857034, 1.16566286808580849780323258043, 2.75602850459425458663151904508, 4.00461985545960213412474245634, 5.44558096460866677181328202197, 6.46728649234214096330961986722, 7.42840002210079633494617822779, 8.118222509118806205369521353713, 8.941572012338263859574986210825, 9.895668854225745190025226420581
