| L(s) = 1 | − 10·3-s − 5·5-s + 18·7-s + 73·9-s + 16·11-s − 6·13-s + 50·15-s − 6·17-s + 124·19-s − 180·21-s − 42·23-s + 25·25-s − 460·27-s + 142·29-s + 188·31-s − 160·33-s − 90·35-s + 202·37-s + 60·39-s + 54·41-s − 66·43-s − 365·45-s − 38·47-s − 19·49-s + 60·51-s + 738·53-s − 80·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 0.447·5-s + 0.971·7-s + 2.70·9-s + 0.438·11-s − 0.128·13-s + 0.860·15-s − 0.0856·17-s + 1.49·19-s − 1.87·21-s − 0.380·23-s + 1/5·25-s − 3.27·27-s + 0.909·29-s + 1.08·31-s − 0.844·33-s − 0.434·35-s + 0.897·37-s + 0.246·39-s + 0.205·41-s − 0.234·43-s − 1.20·45-s − 0.117·47-s − 0.0553·49-s + 0.164·51-s + 1.91·53-s − 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8500871690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8500871690\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 3 | \( 1 + 10 T + p^{3} T^{2} \) |
| 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 - 142 T + p^{3} T^{2} \) |
| 31 | \( 1 - 188 T + p^{3} T^{2} \) |
| 37 | \( 1 - 202 T + p^{3} T^{2} \) |
| 41 | \( 1 - 54 T + p^{3} T^{2} \) |
| 43 | \( 1 + 66 T + p^{3} T^{2} \) |
| 47 | \( 1 + 38 T + p^{3} T^{2} \) |
| 53 | \( 1 - 738 T + p^{3} T^{2} \) |
| 59 | \( 1 + 564 T + p^{3} T^{2} \) |
| 61 | \( 1 + 262 T + p^{3} T^{2} \) |
| 67 | \( 1 - 554 T + p^{3} T^{2} \) |
| 71 | \( 1 + 140 T + p^{3} T^{2} \) |
| 73 | \( 1 - 882 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1160 T + p^{3} T^{2} \) |
| 83 | \( 1 + 642 T + p^{3} T^{2} \) |
| 89 | \( 1 + 854 T + p^{3} T^{2} \) |
| 97 | \( 1 + 478 T + p^{3} T^{2} \) |
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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
−13.78247604844809763172869513363, −12.26661150001202446521892129410, −11.71443659582460772428310846230, −10.94224779586392488463387040264, −9.789317114889372739036717967493, −7.85588036484120296165597145663, −6.66296563216522418263724890126, −5.37388422918337590084021547663, −4.37916787480797941969200020954, −1.02460041804976112977130929889,
1.02460041804976112977130929889, 4.37916787480797941969200020954, 5.37388422918337590084021547663, 6.66296563216522418263724890126, 7.85588036484120296165597145663, 9.789317114889372739036717967493, 10.94224779586392488463387040264, 11.71443659582460772428310846230, 12.26661150001202446521892129410, 13.78247604844809763172869513363
