| L(s) = 1 | − 5-s − 9-s + 25-s + 2·29-s − 2·41-s + 45-s − 49-s − 2·61-s + 81-s + 2·89-s + 2·101-s − 2·109-s + ⋯ |
| L(s) = 1 | − 5-s − 9-s + 25-s + 2·29-s − 2·41-s + 45-s − 49-s − 2·61-s + 81-s + 2·89-s + 2·101-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4643475584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4643475584\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−14.71004762413275441512730195253, −13.69240792583054166179238644349, −12.27030224826286889685238700312, −11.54177841888707105180080511420, −10.41708110873279474671747278381, −8.820916987388348246449123416127, −7.935729000773875944090370436264, −6.49941746742761161543060822888, −4.84156760187899842160701524540, −3.18874463093794795645386180893,
3.18874463093794795645386180893, 4.84156760187899842160701524540, 6.49941746742761161543060822888, 7.935729000773875944090370436264, 8.820916987388348246449123416127, 10.41708110873279474671747278381, 11.54177841888707105180080511420, 12.27030224826286889685238700312, 13.69240792583054166179238644349, 14.71004762413275441512730195253
