| L(s) = 1 | − 3-s + 5-s + 5·7-s − 2·9-s + 2·11-s + 13-s − 15-s − 3·17-s + 2·19-s − 5·21-s + 4·23-s − 4·25-s + 5·27-s + 6·29-s − 4·31-s − 2·33-s + 5·35-s − 11·37-s − 39-s + 8·41-s + 43-s − 2·45-s + 9·47-s + 18·49-s + 3·51-s + 12·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.88·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.727·17-s + 0.458·19-s − 1.09·21-s + 0.834·23-s − 4/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s − 0.348·33-s + 0.845·35-s − 1.80·37-s − 0.160·39-s + 1.24·41-s + 0.152·43-s − 0.298·45-s + 1.31·47-s + 18/7·49-s + 0.420·51-s + 1.64·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.673878937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673878937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−10.61426311245641606337260657814, −9.154616175097776551326098779823, −8.634128369650386182563772881528, −7.68682352791961768327417436418, −6.68813075345386975069351488469, −5.62140548280891007044939038742, −5.06588144648753576824565791164, −4.03445731769668830638718429665, −2.38725239307784062333917973517, −1.20288248237337261583265825731,
1.20288248237337261583265825731, 2.38725239307784062333917973517, 4.03445731769668830638718429665, 5.06588144648753576824565791164, 5.62140548280891007044939038742, 6.68813075345386975069351488469, 7.68682352791961768327417436418, 8.634128369650386182563772881528, 9.154616175097776551326098779823, 10.61426311245641606337260657814
