| L(s) = 1 | + 1.18·2-s + 3-s − 0.606·4-s + 0.961·5-s + 1.18·6-s − 7-s − 3.07·8-s + 9-s + 1.13·10-s + 3.15·11-s − 0.606·12-s − 4.59·13-s − 1.18·14-s + 0.961·15-s − 2.42·16-s + 6.22·17-s + 1.18·18-s + 3.21·19-s − 0.582·20-s − 21-s + 3.72·22-s + 0.430·23-s − 3.07·24-s − 4.07·25-s − 5.42·26-s + 27-s + 0.606·28-s + ⋯ |
| L(s) = 1 | + 0.834·2-s + 0.577·3-s − 0.303·4-s + 0.430·5-s + 0.481·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s + 0.358·10-s + 0.950·11-s − 0.175·12-s − 1.27·13-s − 0.315·14-s + 0.248·15-s − 0.605·16-s + 1.50·17-s + 0.278·18-s + 0.737·19-s − 0.130·20-s − 0.218·21-s + 0.793·22-s + 0.0898·23-s − 0.628·24-s − 0.815·25-s − 1.06·26-s + 0.192·27-s + 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.310665160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.310665160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 5 | \( 1 - 0.961T + 5T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 - 0.430T + 23T^{2} \) |
| 29 | \( 1 - 0.690T + 29T^{2} \) |
| 31 | \( 1 - 5.00T + 31T^{2} \) |
| 37 | \( 1 - 9.34T + 37T^{2} \) |
| 41 | \( 1 + 6.97T + 41T^{2} \) |
| 43 | \( 1 + 0.172T + 43T^{2} \) |
| 47 | \( 1 - 0.252T + 47T^{2} \) |
| 53 | \( 1 + 2.60T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.54T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 + 0.951T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 5.68T + 83T^{2} \) |
| 89 | \( 1 - 6.78T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.452203047903709247888754454907, −7.70510225863276046190528012595, −6.88618788065848018120914933753, −6.07487844513874906048189772694, −5.38383591939666684045281032494, −4.65093724312440519672556906099, −3.77998682727922725432587370531, −3.15795435630993320430644386316, −2.29850334910109877646785864055, −0.917999928888242528369323473832,
0.917999928888242528369323473832, 2.29850334910109877646785864055, 3.15795435630993320430644386316, 3.77998682727922725432587370531, 4.65093724312440519672556906099, 5.38383591939666684045281032494, 6.07487844513874906048189772694, 6.88618788065848018120914933753, 7.70510225863276046190528012595, 8.452203047903709247888754454907
