| L(s) = 1 | − 1.88·2-s − 1.39·3-s + 1.54·4-s − 3.71·5-s + 2.62·6-s + 0.268·7-s + 0.854·8-s − 1.05·9-s + 7.00·10-s + 11-s − 2.15·12-s + 4.11·13-s − 0.504·14-s + 5.18·15-s − 4.70·16-s + 17-s + 1.98·18-s − 7.20·19-s − 5.74·20-s − 0.373·21-s − 1.88·22-s + 5.54·23-s − 1.19·24-s + 8.82·25-s − 7.75·26-s + 5.65·27-s + 0.414·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 0.804·3-s + 0.773·4-s − 1.66·5-s + 1.07·6-s + 0.101·7-s + 0.302·8-s − 0.352·9-s + 2.21·10-s + 0.301·11-s − 0.622·12-s + 1.14·13-s − 0.134·14-s + 1.33·15-s − 1.17·16-s + 0.242·17-s + 0.468·18-s − 1.65·19-s − 1.28·20-s − 0.0815·21-s − 0.401·22-s + 1.15·23-s − 0.243·24-s + 1.76·25-s − 1.52·26-s + 1.08·27-s + 0.0783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4853332452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4853332452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 7 | \( 1 - 0.268T + 7T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 - 5.54T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 2.01T + 41T^{2} \) |
| 47 | \( 1 + 2.88T + 47T^{2} \) |
| 53 | \( 1 - 8.83T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 - 4.31T + 61T^{2} \) |
| 67 | \( 1 + 7.87T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 79 | \( 1 + 7.38T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.109443258902101771591425321354, −7.29319756584630656655860574112, −6.56516616904834886638125945378, −6.13230142511375537318717716384, −4.80164663191796772300271179745, −4.42758237768723223989117203202, −3.54312277997777143438265415877, −2.53703526381654329852014343322, −1.05269734565842975135599400695, −0.57708191489710165487741319584,
0.57708191489710165487741319584, 1.05269734565842975135599400695, 2.53703526381654329852014343322, 3.54312277997777143438265415877, 4.42758237768723223989117203202, 4.80164663191796772300271179745, 6.13230142511375537318717716384, 6.56516616904834886638125945378, 7.29319756584630656655860574112, 8.109443258902101771591425321354
