| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 307.·5-s − 216·6-s + 343·7-s + 512·8-s + 729·9-s − 2.45e3·10-s − 7.42e3·11-s − 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s + 8.29e3·15-s + 4.09e3·16-s − 5.21e3·17-s + 5.83e3·18-s − 610.·19-s − 1.96e4·20-s − 9.26e3·21-s − 5.93e4·22-s − 5.37e4·23-s − 1.38e4·24-s + 1.62e4·25-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 − 1.09·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.777·10-s − 1.68·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.634·15-s + 0.250·16-s − 0.257·17-s + 0.235·18-s − 0.0204·19-s − 0.549·20-s − 0.218·21-s − 1.18·22-s − 0.920·23-s − 0.204·24-s + 0.207·25-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
| \(L(4)\) |
\(\approx\) |
\(1.059846550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059846550\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 307.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 7.42e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 5.21e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 610.T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.54e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.36e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.43e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.84e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.27e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.71e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.16e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.21e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.28e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.74e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.47e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.82e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.84e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.19e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.79e6T + 8.07e13T^{2} \) |
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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
−10.07524407264471751223399208933, −8.432929161817384002756926376178, −7.75696541943460921577405065041, −7.00290512643810163575246176709, −5.78719601609356941786053732132, −4.97084463779556042520767226837, −4.22653887804723747733963001340, −3.11912767045798532668076529501, −1.96464194231332228145152216484, −0.39104131176921821859353698190,
0.39104131176921821859353698190, 1.96464194231332228145152216484, 3.11912767045798532668076529501, 4.22653887804723747733963001340, 4.97084463779556042520767226837, 5.78719601609356941786053732132, 7.00290512643810163575246176709, 7.75696541943460921577405065041, 8.432929161817384002756926376178, 10.07524407264471751223399208933
