| L(s) = 1 | + 0.940·2-s − 3-s − 1.11·4-s − 1.95·5-s − 0.940·6-s − 0.363·7-s − 2.92·8-s + 9-s − 1.83·10-s + 1.99·11-s + 1.11·12-s + 13-s − 0.341·14-s + 1.95·15-s − 0.521·16-s − 2.22·17-s + 0.940·18-s − 5.96·19-s + 2.17·20-s + 0.363·21-s + 1.87·22-s − 1.97·23-s + 2.92·24-s − 1.19·25-s + 0.940·26-s − 27-s + 0.405·28-s + ⋯ |
| L(s) = 1 | + 0.664·2-s − 0.577·3-s − 0.558·4-s − 0.872·5-s − 0.383·6-s − 0.137·7-s − 1.03·8-s + 0.333·9-s − 0.579·10-s + 0.602·11-s + 0.322·12-s + 0.277·13-s − 0.0912·14-s + 0.503·15-s − 0.130·16-s − 0.540·17-s + 0.221·18-s − 1.36·19-s + 0.486·20-s + 0.0792·21-s + 0.400·22-s − 0.412·23-s + 0.597·24-s − 0.238·25-s + 0.184·26-s − 0.192·27-s + 0.0766·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8005955759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8005955759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.940T + 2T^{2} \) |
| 5 | \( 1 + 1.95T + 5T^{2} \) |
| 7 | \( 1 + 0.363T + 7T^{2} \) |
| 11 | \( 1 - 1.99T + 11T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 + 5.96T + 19T^{2} \) |
| 23 | \( 1 + 1.97T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 + 6.18T + 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 + 0.409T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 + 2.75T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 - 0.0721T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 8.03T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 8.54T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.371225919215319820338240989675, −7.77374488034671271826916673456, −6.67426227984074839893874201629, −6.23088387109360969376775883237, −5.38144382192121143317967224229, −4.50461836214089917012519051831, −4.01891579154996419992822372886, −3.39725314425908771579295444503, −2.01495372582180235450289882683, −0.46471390766637537831850624826,
0.46471390766637537831850624826, 2.01495372582180235450289882683, 3.39725314425908771579295444503, 4.01891579154996419992822372886, 4.50461836214089917012519051831, 5.38144382192121143317967224229, 6.23088387109360969376775883237, 6.67426227984074839893874201629, 7.77374488034671271826916673456, 8.371225919215319820338240989675
