| L(s) = 1 | − 3-s + 3·7-s + 9-s − 4·11-s + 13-s − 17-s − 2·19-s − 3·21-s − 27-s − 29-s + 4·33-s − 5·37-s − 39-s + 8·41-s − 43-s + 47-s + 2·49-s + 51-s + 2·57-s − 8·59-s − 4·61-s + 3·63-s − 8·67-s + 6·71-s − 10·73-s − 12·77-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.242·17-s − 0.458·19-s − 0.654·21-s − 0.192·27-s − 0.185·29-s + 0.696·33-s − 0.821·37-s − 0.160·39-s + 1.24·41-s − 0.152·43-s + 0.145·47-s + 2/7·49-s + 0.140·51-s + 0.264·57-s − 1.04·59-s − 0.512·61-s + 0.377·63-s − 0.977·67-s + 0.712·71-s − 1.17·73-s − 1.36·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.33877254885651, −14.05754690671178, −13.31912118871837, −12.97098442513926, −12.46268392648843, −11.80294761064312, −11.45980137645532, −10.83894289933654, −10.51539716245435, −10.18906431029308, −9.160362476780267, −8.951684702780783, −8.084387747729176, −7.799257230973603, −7.335953769251613, −6.558922949982124, −5.994855474267173, −5.496348859261997, −4.782494594508802, −4.655528112708122, −3.796160165640166, −3.054303696511818, −2.231927124463336, −1.751448565811003, −0.8745301558921360, 0,
0.8745301558921360, 1.751448565811003, 2.231927124463336, 3.054303696511818, 3.796160165640166, 4.655528112708122, 4.782494594508802, 5.496348859261997, 5.994855474267173, 6.558922949982124, 7.335953769251613, 7.799257230973603, 8.084387747729176, 8.951684702780783, 9.160362476780267, 10.18906431029308, 10.51539716245435, 10.83894289933654, 11.45980137645532, 11.80294761064312, 12.46268392648843, 12.97098442513926, 13.31912118871837, 14.05754690671178, 14.33877254885651
