| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 11-s − 2·13-s − 15-s − 8·17-s + 2·19-s − 2·21-s − 3·23-s + 25-s − 27-s − 29-s − 4·31-s + 33-s + 2·35-s − 11·37-s + 2·39-s + 9·41-s + 9·43-s + 45-s + 10·47-s − 3·49-s + 8·51-s + 3·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.258·15-s − 1.94·17-s + 0.458·19-s − 0.436·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s − 0.718·31-s + 0.174·33-s + 0.338·35-s − 1.80·37-s + 0.320·39-s + 1.40·41-s + 1.37·43-s + 0.149·45-s + 1.45·47-s − 3/7·49-s + 1.12·51-s + 0.412·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526764862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526764862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| 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
−7.77587194669958269438501651031, −7.25879547872953062310545423718, −6.54013350233004086520448341245, −5.75422476391767050887755612300, −5.15982584316045413214787432487, −4.51223031675412813471347153552, −3.76309085572759877548067081133, −2.38494335172971112956137725776, −1.96219875656599317733420927846, −0.63311432603305038135816002960,
0.63311432603305038135816002960, 1.96219875656599317733420927846, 2.38494335172971112956137725776, 3.76309085572759877548067081133, 4.51223031675412813471347153552, 5.15982584316045413214787432487, 5.75422476391767050887755612300, 6.54013350233004086520448341245, 7.25879547872953062310545423718, 7.77587194669958269438501651031
