| L(s) = 1 | − 2·4-s − 5-s + 2·7-s + 3·11-s + 4·13-s + 4·16-s + 2·20-s + 6·23-s + 25-s − 4·28-s + 3·29-s − 5·31-s − 2·35-s − 8·37-s − 6·41-s − 4·43-s − 6·44-s − 6·47-s − 3·49-s − 8·52-s − 6·53-s − 3·55-s − 9·59-s − 7·61-s − 8·64-s − 4·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.755·7-s + 0.904·11-s + 1.10·13-s + 16-s + 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.755·28-s + 0.557·29-s − 0.898·31-s − 0.338·35-s − 1.31·37-s − 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.875·47-s − 3/7·49-s − 1.10·52-s − 0.824·53-s − 0.404·55-s − 1.17·59-s − 0.896·61-s − 64-s − 0.496·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.24829612810809, −15.63334310445578, −14.86813195076504, −14.68151000234201, −13.96789764479291, −13.52058928395821, −12.99253657647404, −12.27989656472158, −11.80573222586068, −11.13023482981508, −10.70603162170148, −10.00126661624895, −9.066072187159856, −8.957344399867594, −8.277149390922542, −7.777426039744632, −6.939666790693225, −6.342751220658681, −5.516633157935133, −4.815609230056924, −4.453717839069794, −3.462008683487843, −3.293612713999307, −1.660432324832200, −1.206162012622432, 0,
1.206162012622432, 1.660432324832200, 3.293612713999307, 3.462008683487843, 4.453717839069794, 4.815609230056924, 5.516633157935133, 6.342751220658681, 6.939666790693225, 7.777426039744632, 8.277149390922542, 8.957344399867594, 9.066072187159856, 10.00126661624895, 10.70603162170148, 11.13023482981508, 11.80573222586068, 12.27989656472158, 12.99253657647404, 13.52058928395821, 13.96789764479291, 14.68151000234201, 14.86813195076504, 15.63334310445578, 16.24829612810809
