L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s + 13-s − 17-s + 4·19-s + 2·21-s + 2·23-s − 5·25-s − 27-s + 4·29-s + 2·31-s + 4·33-s − 39-s + 2·41-s + 4·43-s + 4·47-s − 3·49-s + 51-s − 6·53-s − 4·57-s − 4·59-s − 4·61-s − 2·63-s + 4·67-s − 2·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.436·21-s + 0.417·23-s − 25-s − 0.192·27-s + 0.742·29-s + 0.359·31-s + 0.696·33-s − 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.512·61-s − 0.251·63-s + 0.488·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.26964877486064, −14.30019922582835, −13.86964711956048, −13.36991840905936, −12.82665593563599, −12.51608771152242, −11.79079317259199, −11.39676565375899, −10.68304580962872, −10.39726959556540, −9.674982420725369, −9.409227125656228, −8.592264432473711, −7.948403957311793, −7.493466121739667, −6.883958139797551, −6.236061900490314, −5.784470371748757, −5.206626648530351, −4.604241227169729, −3.926460878518254, −3.092754889123018, −2.698957658352503, −1.733127143252445, −0.8066733907338897, 0,
0.8066733907338897, 1.733127143252445, 2.698957658352503, 3.092754889123018, 3.926460878518254, 4.604241227169729, 5.206626648530351, 5.784470371748757, 6.236061900490314, 6.883958139797551, 7.493466121739667, 7.948403957311793, 8.592264432473711, 9.409227125656228, 9.674982420725369, 10.39726959556540, 10.68304580962872, 11.39676565375899, 11.79079317259199, 12.51608771152242, 12.82665593563599, 13.36991840905936, 13.86964711956048, 14.30019922582835, 15.26964877486064