| L(s) = 1 | + 3-s + 4·7-s + 9-s − 11-s − 13-s + 2·17-s + 4·19-s + 4·21-s + 4·23-s + 27-s + 6·29-s − 33-s − 10·37-s − 39-s − 10·41-s − 4·43-s + 9·49-s + 2·51-s − 10·53-s + 4·57-s − 12·59-s − 2·61-s + 4·63-s − 8·67-s + 4·69-s − 8·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s + 0.872·21-s + 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.174·33-s − 1.64·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s + 9/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + 0.503·63-s − 0.977·67-s + 0.481·69-s − 0.949·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.07585458716458, −13.85340950778924, −13.35668498657162, −12.64061570614340, −12.09143292278254, −11.79495165046882, −11.20379682256527, −10.64977800503360, −10.19922601861682, −9.737604492303445, −8.944002144849675, −8.642689230155592, −8.129417464123340, −7.594252445483105, −7.267605792824229, −6.615845851882909, −5.856165850482684, −5.078829687536189, −4.918172162947348, −4.394131941480071, −3.318780480169980, −3.180817371101257, −2.301013889351919, −1.501163918793157, −1.278685302024058, 0,
1.278685302024058, 1.501163918793157, 2.301013889351919, 3.180817371101257, 3.318780480169980, 4.394131941480071, 4.918172162947348, 5.078829687536189, 5.856165850482684, 6.615845851882909, 7.267605792824229, 7.594252445483105, 8.129417464123340, 8.642689230155592, 8.944002144849675, 9.737604492303445, 10.19922601861682, 10.64977800503360, 11.20379682256527, 11.79495165046882, 12.09143292278254, 12.64061570614340, 13.35668498657162, 13.85340950778924, 14.07585458716458
