| L(s) = 1 | − 4.11·2-s − 111.·4-s + 125·5-s + 343·7-s + 983.·8-s − 514.·10-s + 4.66e3·11-s + 5.47e3·13-s − 1.41e3·14-s + 1.01e4·16-s − 3.72e3·17-s + 4.79e4·19-s − 1.38e4·20-s − 1.91e4·22-s − 5.85e4·23-s + 1.56e4·25-s − 2.25e4·26-s − 3.81e4·28-s + 1.90e5·29-s − 9.82e4·31-s − 1.67e5·32-s + 1.53e4·34-s + 4.28e4·35-s + 2.00e5·37-s − 1.97e5·38-s + 1.22e5·40-s + 2.31e4·41-s + ⋯ |
| L(s) = 1 | − 0.363·2-s − 0.867·4-s + 0.447·5-s + 0.377·7-s + 0.679·8-s − 0.162·10-s + 1.05·11-s + 0.691·13-s − 0.137·14-s + 0.620·16-s − 0.183·17-s + 1.60·19-s − 0.388·20-s − 0.383·22-s − 1.00·23-s + 0.199·25-s − 0.251·26-s − 0.328·28-s + 1.45·29-s − 0.592·31-s − 0.904·32-s + 0.0668·34-s + 0.169·35-s + 0.650·37-s − 0.583·38-s + 0.303·40-s + 0.0525·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.992098398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992098398\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 + 4.11T + 128T^{2} \) |
| 11 | \( 1 - 4.66e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.47e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.72e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.90e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.82e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.31e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.84e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.89e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.32e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.88e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.85e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.66e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.59e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.31e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.35e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.99e6T + 8.07e13T^{2} \) |
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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
−10.17547483737108297349704152275, −9.492197613928703324715672628767, −8.671860166330310621657717487555, −7.83064894983703769329473399609, −6.56696323922347938014832194039, −5.47457782827690615471708842201, −4.44236708460964375912492680876, −3.38741228925464854287449271912, −1.64340069742392157321831715935, −0.796405820617826687647526462390,
0.796405820617826687647526462390, 1.64340069742392157321831715935, 3.38741228925464854287449271912, 4.44236708460964375912492680876, 5.47457782827690615471708842201, 6.56696323922347938014832194039, 7.83064894983703769329473399609, 8.671860166330310621657717487555, 9.492197613928703324715672628767, 10.17547483737108297349704152275
