| L(s) = 1 | − 7.49·2-s − 71.8·4-s − 400.·5-s − 272.·7-s + 1.49e3·8-s + 3.00e3·10-s − 1.28e3·11-s + 9.53e3·13-s + 2.03e3·14-s − 2.02e3·16-s + 3.54e4·17-s − 1.44e4·19-s + 2.87e4·20-s + 9.64e3·22-s − 5.12e4·23-s + 8.25e4·25-s − 7.14e4·26-s + 1.95e4·28-s + 7.99e4·29-s + 2.28e5·31-s − 1.76e5·32-s − 2.65e5·34-s + 1.09e5·35-s − 1.39e5·37-s + 1.08e5·38-s − 6.00e5·40-s − 4.97e5·41-s + ⋯ |
| L(s) = 1 | − 0.662·2-s − 0.561·4-s − 1.43·5-s − 0.299·7-s + 1.03·8-s + 0.949·10-s − 0.291·11-s + 1.20·13-s + 0.198·14-s − 0.123·16-s + 1.75·17-s − 0.482·19-s + 0.804·20-s + 0.193·22-s − 0.878·23-s + 1.05·25-s − 0.797·26-s + 0.168·28-s + 0.608·29-s + 1.37·31-s − 0.952·32-s − 1.16·34-s + 0.430·35-s − 0.454·37-s + 0.319·38-s − 1.48·40-s − 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6779878975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6779878975\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 7.49T + 128T^{2} \) |
| 5 | \( 1 + 400.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 272.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.28e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.53e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.54e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.44e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.12e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.99e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.28e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.97e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.80e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.78e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.36e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.85e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.58e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.27e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.92e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.34e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.12e6T + 8.07e13T^{2} \) |
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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
−9.783189833621504970129500181238, −8.383981334051938249734390241014, −8.287217720115549744445561814549, −7.37452665124491574883096179462, −6.15954884871177047339227280427, −4.89104135429305019123255048520, −3.93048642217529969505416652807, −3.23356609874796441196962860674, −1.35442011602447113441428845045, −0.44615554965021455828626538137,
0.44615554965021455828626538137, 1.35442011602447113441428845045, 3.23356609874796441196962860674, 3.93048642217529969505416652807, 4.89104135429305019123255048520, 6.15954884871177047339227280427, 7.37452665124491574883096179462, 8.287217720115549744445561814549, 8.383981334051938249734390241014, 9.783189833621504970129500181238
