| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s − 525.·5-s + 216·6-s − 343·7-s − 512·8-s + 729·9-s + 4.20e3·10-s + 6.26e3·11-s − 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s + 1.41e4·15-s + 4.09e3·16-s + 3.45e4·17-s − 5.83e3·18-s + 1.78e3·19-s − 3.36e4·20-s + 9.26e3·21-s − 5.01e4·22-s − 4.53e4·23-s + 1.38e4·24-s + 1.97e5·25-s + 1.75e4·26-s − 1.96e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.87·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.32·10-s + 1.41·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1.08·15-s + 0.250·16-s + 1.70·17-s − 0.235·18-s + 0.0596·19-s − 0.939·20-s + 0.218·21-s − 1.00·22-s − 0.776·23-s + 0.204·24-s + 2.52·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7716448112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7716448112\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 + 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 5 | \( 1 + 525.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 6.26e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.78e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.53e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.87e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.80e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.12e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.00e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.36e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.53e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 7.04e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.51e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.57e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.02e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.33e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.32e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.39e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.60e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.43e6T + 8.07e13T^{2} \) |
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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
−9.807202540870178733892928184631, −8.558984676237626256979625157663, −7.994166564546857379445542850913, −7.01185802654409186215330072725, −6.42144828366427467425202228838, −4.96083328700772914590288502908, −3.87840372661798260869604782660, −3.16445336061070503374841073099, −1.26710833675637578414850663806, −0.50499999278384923011521851741,
0.50499999278384923011521851741, 1.26710833675637578414850663806, 3.16445336061070503374841073099, 3.87840372661798260869604782660, 4.96083328700772914590288502908, 6.42144828366427467425202228838, 7.01185802654409186215330072725, 7.994166564546857379445542850913, 8.558984676237626256979625157663, 9.807202540870178733892928184631
