| L(s) = 1 | − 2-s + 3-s + 4-s − 4.41·5-s − 6-s + 2·7-s − 8-s + 9-s + 4.41·10-s − 11-s + 12-s + 3·13-s − 2·14-s − 4.41·15-s + 16-s + 2.82·17-s − 18-s + 1.17·19-s − 4.41·20-s + 2·21-s + 22-s − 2.82·23-s − 24-s + 14.4·25-s − 3·26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.97·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.333·9-s + 1.39·10-s − 0.301·11-s + 0.288·12-s + 0.832·13-s − 0.534·14-s − 1.13·15-s + 0.250·16-s + 0.685·17-s − 0.235·18-s + 0.268·19-s − 0.987·20-s + 0.436·21-s + 0.213·22-s − 0.589·23-s − 0.204·24-s + 2.89·25-s − 0.588·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174418615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174418615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + 4.41T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 1.65T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 5.31T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.314291976578453610425888977880, −8.078760550304210497098668502497, −7.20189899092246941679534621455, −6.70362464417814807820538617865, −5.32582134653663642260418831195, −4.53536886038287770044526025115, −3.58842894662310630317536590998, −3.18142311305950756207056981513, −1.78513181048759637198992656317, −0.68512084179758343425646155118,
0.68512084179758343425646155118, 1.78513181048759637198992656317, 3.18142311305950756207056981513, 3.58842894662310630317536590998, 4.53536886038287770044526025115, 5.32582134653663642260418831195, 6.70362464417814807820538617865, 7.20189899092246941679534621455, 8.078760550304210497098668502497, 8.314291976578453610425888977880
