| L(s) = 1 | − 0.732·3-s + 5-s + 2·7-s − 2.46·9-s + 3.46·11-s + 0.732·13-s − 0.732·15-s + 3.46·17-s + 19-s − 1.46·21-s + 3.46·23-s + 25-s + 4·27-s − 3.46·29-s + 5.46·31-s − 2.53·33-s + 2·35-s + 3.26·37-s − 0.535·39-s − 6·41-s + 8.92·43-s − 2.46·45-s − 0.928·47-s − 3·49-s − 2.53·51-s − 7.26·53-s + 3.46·55-s + ⋯ |
| L(s) = 1 | − 0.422·3-s + 0.447·5-s + 0.755·7-s − 0.821·9-s + 1.04·11-s + 0.203·13-s − 0.189·15-s + 0.840·17-s + 0.229·19-s − 0.319·21-s + 0.722·23-s + 0.200·25-s + 0.769·27-s − 0.643·29-s + 0.981·31-s − 0.441·33-s + 0.338·35-s + 0.537·37-s − 0.0858·39-s − 0.937·41-s + 1.36·43-s − 0.367·45-s − 0.135·47-s − 0.428·49-s − 0.355·51-s − 0.998·53-s + 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.394063971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394063971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 0.928T + 47T^{2} \) |
| 53 | \( 1 + 7.26T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 - 3.26T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.44331589690108080271688966417, −10.59458783679243013300771317226, −9.477293355440750799891256755494, −8.665256492777781791062861870619, −7.63467799182691072362611218413, −6.40148018371616747176468977557, −5.60739079263632553289990038191, −4.56237902716627680960834560191, −3.07160637666307823419567404448, −1.36351097618979905690069751548,
1.36351097618979905690069751548, 3.07160637666307823419567404448, 4.56237902716627680960834560191, 5.60739079263632553289990038191, 6.40148018371616747176468977557, 7.63467799182691072362611218413, 8.665256492777781791062861870619, 9.477293355440750799891256755494, 10.59458783679243013300771317226, 11.44331589690108080271688966417
