| L(s) = 1 | + 0.0134·3-s + 3.34·5-s − 3.13·7-s − 2.99·9-s + 4.36·11-s − 4.11·13-s + 0.0449·15-s + 17-s − 8.39·19-s − 0.0421·21-s + 0.00124·23-s + 6.20·25-s − 0.0805·27-s − 0.781·29-s + 9.27·31-s + 0.0586·33-s − 10.4·35-s + 2.20·37-s − 0.0552·39-s + 11.7·41-s + 10.9·43-s − 10.0·45-s + 13.3·47-s + 2.83·49-s + 0.0134·51-s + 0.279·53-s + 14.6·55-s + ⋯ |
| L(s) = 1 | + 0.00775·3-s + 1.49·5-s − 1.18·7-s − 0.999·9-s + 1.31·11-s − 1.14·13-s + 0.0116·15-s + 0.242·17-s − 1.92·19-s − 0.00918·21-s + 0.000260·23-s + 1.24·25-s − 0.0155·27-s − 0.145·29-s + 1.66·31-s + 0.0102·33-s − 1.77·35-s + 0.363·37-s − 0.00884·39-s + 1.83·41-s + 1.66·43-s − 1.49·45-s + 1.94·47-s + 0.405·49-s + 0.00187·51-s + 0.0384·53-s + 1.97·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.964026344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.964026344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0134T + 3T^{2} \) |
| 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 + 4.11T + 13T^{2} \) |
| 19 | \( 1 + 8.39T + 19T^{2} \) |
| 23 | \( 1 - 0.00124T + 23T^{2} \) |
| 29 | \( 1 + 0.781T + 29T^{2} \) |
| 31 | \( 1 - 9.27T + 31T^{2} \) |
| 37 | \( 1 - 2.20T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 0.279T + 53T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 4.73T + 67T^{2} \) |
| 71 | \( 1 - 9.56T + 71T^{2} \) |
| 73 | \( 1 - 4.22T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 0.0627T + 83T^{2} \) |
| 89 | \( 1 - 9.62T + 89T^{2} \) |
| 97 | \( 1 - 4.12T + 97T^{2} \) |
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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
−8.757584704661239040470164827021, −7.67420177679666976125319810927, −6.62062009801049640629646955927, −6.20276919620638522710797167232, −5.83076696240864218607654316488, −4.70207261795969479834623411135, −3.83188624617337421402260228272, −2.58340705760419313039010814684, −2.32807703653082327583608961561, −0.78111444375351440949165780513,
0.78111444375351440949165780513, 2.32807703653082327583608961561, 2.58340705760419313039010814684, 3.83188624617337421402260228272, 4.70207261795969479834623411135, 5.83076696240864218607654316488, 6.20276919620638522710797167232, 6.62062009801049640629646955927, 7.67420177679666976125319810927, 8.757584704661239040470164827021
