| L(s) = 1 | − 1.79·2-s − 3-s + 1.23·4-s + 1.64·5-s + 1.79·6-s + 1.60·7-s + 1.37·8-s + 9-s − 2.96·10-s − 1.34·11-s − 1.23·12-s + 13-s − 2.89·14-s − 1.64·15-s − 4.94·16-s − 5.68·17-s − 1.79·18-s − 1.69·19-s + 2.03·20-s − 1.60·21-s + 2.42·22-s − 1.20·23-s − 1.37·24-s − 2.28·25-s − 1.79·26-s − 27-s + 1.99·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.618·4-s + 0.736·5-s + 0.734·6-s + 0.608·7-s + 0.485·8-s + 0.333·9-s − 0.937·10-s − 0.406·11-s − 0.357·12-s + 0.277·13-s − 0.774·14-s − 0.425·15-s − 1.23·16-s − 1.37·17-s − 0.424·18-s − 0.388·19-s + 0.455·20-s − 0.351·21-s + 0.517·22-s − 0.250·23-s − 0.280·24-s − 0.457·25-s − 0.352·26-s − 0.192·27-s + 0.376·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7405981032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7405981032\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 + 9.37T + 31T^{2} \) |
| 37 | \( 1 + 0.916T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 4.44T + 47T^{2} \) |
| 53 | \( 1 + 6.29T + 53T^{2} \) |
| 59 | \( 1 + 0.514T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 3.26T + 67T^{2} \) |
| 71 | \( 1 - 8.99T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 - 0.847T + 83T^{2} \) |
| 89 | \( 1 + 1.04T + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−8.601452111354309393753227394045, −7.79054673241931382396683601271, −7.17761115383874095133320615465, −6.32482775624907100869511275812, −5.63317256990373997432950853411, −4.73390874610790048494524601453, −4.04486451234456195678271611784, −2.35880467170253295499071349348, −1.79002211825901027412872549772, −0.61932687048761507787696553104,
0.61932687048761507787696553104, 1.79002211825901027412872549772, 2.35880467170253295499071349348, 4.04486451234456195678271611784, 4.73390874610790048494524601453, 5.63317256990373997432950853411, 6.32482775624907100869511275812, 7.17761115383874095133320615465, 7.79054673241931382396683601271, 8.601452111354309393753227394045
