| L(s) = 1 | − 0.0447·2-s − 2.38·3-s − 1.99·4-s − 5-s + 0.106·6-s − 7-s + 0.179·8-s + 2.69·9-s + 0.0447·10-s + 1.58·11-s + 4.76·12-s + 4.30·13-s + 0.0447·14-s + 2.38·15-s + 3.98·16-s − 0.131·17-s − 0.120·18-s + 5.31·19-s + 1.99·20-s + 2.38·21-s − 0.0708·22-s + 4.65·23-s − 0.427·24-s + 25-s − 0.192·26-s + 0.717·27-s + 1.99·28-s + ⋯ |
| L(s) = 1 | − 0.0316·2-s − 1.37·3-s − 0.998·4-s − 0.447·5-s + 0.0436·6-s − 0.377·7-s + 0.0633·8-s + 0.899·9-s + 0.0141·10-s + 0.477·11-s + 1.37·12-s + 1.19·13-s + 0.0119·14-s + 0.616·15-s + 0.996·16-s − 0.0319·17-s − 0.0285·18-s + 1.22·19-s + 0.446·20-s + 0.520·21-s − 0.0151·22-s + 0.969·23-s − 0.0872·24-s + 0.200·25-s − 0.0377·26-s + 0.138·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.056318468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.056318468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0447T + 2T^{2} \) |
| 3 | \( 1 + 2.38T + 3T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 0.131T + 17T^{2} \) |
| 19 | \( 1 - 5.31T + 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 - 7.13T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 2.91T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 - 5.61T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 0.726T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−7.921910602346235144202503648502, −6.86075057248772057546479527264, −6.46072257844527912999311716719, −5.61127169260964290860721607575, −5.15749118617502466996446262966, −4.36849820962136695229097530366, −3.74971221423512881529908406193, −2.90712570701496996284574924088, −1.03760939309099533358431757693, −0.76501296622650206961747589944,
0.76501296622650206961747589944, 1.03760939309099533358431757693, 2.90712570701496996284574924088, 3.74971221423512881529908406193, 4.36849820962136695229097530366, 5.15749118617502466996446262966, 5.61127169260964290860721607575, 6.46072257844527912999311716719, 6.86075057248772057546479527264, 7.921910602346235144202503648502
