| L(s) = 1 | − 2-s − 1.61·3-s + 4-s − 5-s + 1.61·6-s + 7-s − 8-s − 0.381·9-s + 10-s − 1.61·12-s − 0.763·13-s − 14-s + 1.61·15-s + 16-s + 0.618·17-s + 0.381·18-s − 2.61·19-s − 20-s − 1.61·21-s + 0.472·23-s + 1.61·24-s + 25-s + 0.763·26-s + 5.47·27-s + 28-s + 4·29-s − 1.61·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.934·3-s + 0.5·4-s − 0.447·5-s + 0.660·6-s + 0.377·7-s − 0.353·8-s − 0.127·9-s + 0.316·10-s − 0.467·12-s − 0.211·13-s − 0.267·14-s + 0.417·15-s + 0.250·16-s + 0.149·17-s + 0.0900·18-s − 0.600·19-s − 0.223·20-s − 0.353·21-s + 0.0984·23-s + 0.330·24-s + 0.200·25-s + 0.149·26-s + 1.05·27-s + 0.188·28-s + 0.742·29-s − 0.295·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6119664637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6119664637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 - 0.472T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 - 6.38T + 41T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 0.0901T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.78433942543069273731384899382, −7.16085862759889654441995851646, −6.36565943703155767575419982657, −5.94523920042829001315141462800, −4.96036251354591924905864099665, −4.52213184021877608414672465847, −3.37368447540678475986188033663, −2.55700424230590198328496325820, −1.45370987193712808823075142134, −0.46805817181426571623948303337,
0.46805817181426571623948303337, 1.45370987193712808823075142134, 2.55700424230590198328496325820, 3.37368447540678475986188033663, 4.52213184021877608414672465847, 4.96036251354591924905864099665, 5.94523920042829001315141462800, 6.36565943703155767575419982657, 7.16085862759889654441995851646, 7.78433942543069273731384899382
