| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 6·11-s + 4·13-s − 15-s − 2·17-s + 19-s − 2·21-s + 4·23-s + 25-s − 27-s − 8·31-s − 6·33-s + 2·35-s + 8·37-s − 4·39-s − 2·43-s + 45-s + 4·47-s − 3·49-s + 2·51-s − 14·53-s + 6·55-s − 57-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s − 0.258·15-s − 0.485·17-s + 0.229·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.43·31-s − 1.04·33-s + 0.338·35-s + 1.31·37-s − 0.640·39-s − 0.304·43-s + 0.149·45-s + 0.583·47-s − 3/7·49-s + 0.280·51-s − 1.92·53-s + 0.809·55-s − 0.132·57-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.134716439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.134716439\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
| 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
−9.184809576486902603860883170875, −8.337355231878352341914157738346, −7.35870019741368879643811536078, −6.48306399188922432270388202370, −6.05056538965456241714644666628, −5.04149604884334228319967737624, −4.25458420597759577521531817930, −3.38518004834783914635450255196, −1.81662391816304045133377298599, −1.10256520769340936628415975314,
1.10256520769340936628415975314, 1.81662391816304045133377298599, 3.38518004834783914635450255196, 4.25458420597759577521531817930, 5.04149604884334228319967737624, 6.05056538965456241714644666628, 6.48306399188922432270388202370, 7.35870019741368879643811536078, 8.337355231878352341914157738346, 9.184809576486902603860883170875
