| L(s) = 1 | + 9·3-s + 60.7·5-s − 106.·7-s + 81·9-s + 85.7·11-s + 144.·13-s + 546.·15-s − 989.·17-s − 2.81e3·19-s − 961.·21-s + 529·23-s + 561.·25-s + 729·27-s − 5.97e3·29-s + 7.81e3·31-s + 771.·33-s − 6.48e3·35-s − 1.04e4·37-s + 1.29e3·39-s + 7.75e3·41-s + 7.94e3·43-s + 4.91e3·45-s − 1.51e4·47-s − 5.39e3·49-s − 8.90e3·51-s − 3.04e4·53-s + 5.20e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.08·5-s − 0.824·7-s + 0.333·9-s + 0.213·11-s + 0.237·13-s + 0.627·15-s − 0.830·17-s − 1.78·19-s − 0.475·21-s + 0.208·23-s + 0.179·25-s + 0.192·27-s − 1.31·29-s + 1.46·31-s + 0.123·33-s − 0.895·35-s − 1.25·37-s + 0.136·39-s + 0.720·41-s + 0.655·43-s + 0.362·45-s − 1.00·47-s − 0.320·49-s − 0.479·51-s − 1.49·53-s + 0.232·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 - 60.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 106.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 85.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 144.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 989.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.81e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 5.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.04e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.75e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.94e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 331.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.49e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.87e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.52e5T + 8.58e9T^{2} \) |
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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
−9.408805271787494384095845273386, −8.947508020592408386959138294015, −7.87021820756518535161235379641, −6.49791442843937088061660906305, −6.24407475252015538040564394803, −4.80074151211100634787514854745, −3.67729072229139103923547008810, −2.51335766949966048085329146421, −1.66863553931169778783945113981, 0,
1.66863553931169778783945113981, 2.51335766949966048085329146421, 3.67729072229139103923547008810, 4.80074151211100634787514854745, 6.24407475252015538040564394803, 6.49791442843937088061660906305, 7.87021820756518535161235379641, 8.947508020592408386959138294015, 9.408805271787494384095845273386
