| L(s) = 1 | + 5-s + 25-s − 2·29-s + 2·41-s − 49-s − 2·61-s − 2·89-s − 2·101-s − 2·109-s + ⋯ |
| L(s) = 1 | + 5-s + 25-s − 2·29-s + 2·41-s − 49-s − 2·61-s − 2·89-s − 2·101-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.100269086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.100269086\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 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
−10.65393845174372534363834628703, −9.576965669747807839813673320093, −9.198730912360012737135715031000, −8.023579868542236758654758581506, −7.08634379863268135930907075849, −6.07173207773908522030856944187, −5.39183877784557644532980980512, −4.21033117897709809427824631381, −2.87184327636628604714433286324, −1.67160944678172248872900546019,
1.67160944678172248872900546019, 2.87184327636628604714433286324, 4.21033117897709809427824631381, 5.39183877784557644532980980512, 6.07173207773908522030856944187, 7.08634379863268135930907075849, 8.023579868542236758654758581506, 9.198730912360012737135715031000, 9.576965669747807839813673320093, 10.65393845174372534363834628703
