| L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s − 13-s + 15-s − 19-s + 3·21-s − 6·23-s + 25-s − 27-s + 2·29-s − 3·31-s + 3·35-s − 2·37-s + 39-s − 4·41-s + 43-s − 45-s + 47-s + 2·49-s − 4·53-s + 57-s + 11·59-s + 11·61-s − 3·63-s + 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.229·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.538·31-s + 0.507·35-s − 0.328·37-s + 0.160·39-s − 0.624·41-s + 0.152·43-s − 0.149·45-s + 0.145·47-s + 2/7·49-s − 0.549·53-s + 0.132·57-s + 1.43·59-s + 1.40·61-s − 0.377·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3396981015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3396981015\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
| 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
−13.06044050351137, −12.57919358354541, −12.20925036382540, −11.74007219037706, −11.33552360414878, −10.75010644152739, −10.22080718845644, −9.877540168889876, −9.523467251103411, −8.766289340883949, −8.407827757580270, −7.792945185932653, −7.242621533591805, −6.746707609843142, −6.453886737809645, −5.743149665261016, −5.453535306541840, −4.708160032710058, −4.162775601695028, −3.686587152814751, −3.179247258039129, −2.465753644610962, −1.867221325931332, −0.9938901625171052, −0.1970800521101621,
0.1970800521101621, 0.9938901625171052, 1.867221325931332, 2.465753644610962, 3.179247258039129, 3.686587152814751, 4.162775601695028, 4.708160032710058, 5.453535306541840, 5.743149665261016, 6.453886737809645, 6.746707609843142, 7.242621533591805, 7.792945185932653, 8.407827757580270, 8.766289340883949, 9.523467251103411, 9.877540168889876, 10.22080718845644, 10.75010644152739, 11.33552360414878, 11.74007219037706, 12.20925036382540, 12.57919358354541, 13.06044050351137
