| L(s) = 1 | − 3-s − 4·7-s + 9-s − 5·11-s − 13-s − 17-s + 7·19-s + 4·21-s − 6·23-s − 27-s + 4·29-s + 9·31-s + 5·33-s − 5·37-s + 39-s − 2·41-s + 43-s − 12·47-s + 9·49-s + 51-s − 7·57-s − 4·59-s + 11·61-s − 4·63-s − 10·67-s + 6·69-s + 14·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.242·17-s + 1.60·19-s + 0.872·21-s − 1.25·23-s − 0.192·27-s + 0.742·29-s + 1.61·31-s + 0.870·33-s − 0.821·37-s + 0.160·39-s − 0.312·41-s + 0.152·43-s − 1.75·47-s + 9/7·49-s + 0.140·51-s − 0.927·57-s − 0.520·59-s + 1.40·61-s − 0.503·63-s − 1.22·67-s + 0.722·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6967923391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6967923391\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.77613874500216, −12.33301902915740, −11.92158678730597, −11.54737575156571, −10.91149603474086, −10.29071762921392, −10.03872029828652, −9.776576866592645, −9.332791227574899, −8.381519608772940, −8.250519507196841, −7.530835795683761, −7.121128077013740, −6.582448612825585, −6.189097988900124, −5.663748091204157, −5.181439126259493, −4.751432030640318, −4.119417295387662, −3.318735439446702, −3.079620351546805, −2.509245981534767, −1.814243720822885, −0.8764974004468950, −0.2895593487033456,
0.2895593487033456, 0.8764974004468950, 1.814243720822885, 2.509245981534767, 3.079620351546805, 3.318735439446702, 4.119417295387662, 4.751432030640318, 5.181439126259493, 5.663748091204157, 6.189097988900124, 6.582448612825585, 7.121128077013740, 7.530835795683761, 8.250519507196841, 8.381519608772940, 9.332791227574899, 9.776576866592645, 10.03872029828652, 10.29071762921392, 10.91149603474086, 11.54737575156571, 11.92158678730597, 12.33301902915740, 12.77613874500216
