| L(s) = 1 | − 1.15·2-s − 0.667·4-s + 3.44·5-s + 4.51·7-s + 3.07·8-s − 3.98·10-s + 0.554·11-s + 2.75·13-s − 5.21·14-s − 2.21·16-s + 2.99·17-s − 0.278·19-s − 2.30·20-s − 0.639·22-s − 0.927·23-s + 6.89·25-s − 3.18·26-s − 3.01·28-s + 9.12·31-s − 3.59·32-s − 3.45·34-s + 15.5·35-s − 6.87·37-s + 0.321·38-s + 10.6·40-s + 2.85·41-s − 11.2·43-s + ⋯ |
| L(s) = 1 | − 0.816·2-s − 0.333·4-s + 1.54·5-s + 1.70·7-s + 1.08·8-s − 1.25·10-s + 0.167·11-s + 0.765·13-s − 1.39·14-s − 0.554·16-s + 0.726·17-s − 0.0638·19-s − 0.514·20-s − 0.136·22-s − 0.193·23-s + 1.37·25-s − 0.624·26-s − 0.569·28-s + 1.63·31-s − 0.635·32-s − 0.593·34-s + 2.63·35-s − 1.13·37-s + 0.0520·38-s + 1.67·40-s + 0.445·41-s − 1.71·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.386514836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.386514836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 1.15T + 2T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 11 | \( 1 - 0.554T + 11T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 + 0.278T + 19T^{2} \) |
| 23 | \( 1 + 0.927T + 23T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 4.19T + 47T^{2} \) |
| 53 | \( 1 - 2.07T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 + 0.913T + 73T^{2} \) |
| 79 | \( 1 - 2.06T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 6.80T + 89T^{2} \) |
| 97 | \( 1 + 3.51T + 97T^{2} \) |
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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
−8.156776127075500786389965093130, −7.40066314358144070447698494444, −6.49077659362897102245790593901, −5.75036739227558090743025034871, −5.03670189713552388540652349546, −4.62112533165619204994759912178, −3.51963730269687316593241819313, −2.21897371772611866937310066631, −1.56890303528569594701398545878, −1.01543328522474359073883645238,
1.01543328522474359073883645238, 1.56890303528569594701398545878, 2.21897371772611866937310066631, 3.51963730269687316593241819313, 4.62112533165619204994759912178, 5.03670189713552388540652349546, 5.75036739227558090743025034871, 6.49077659362897102245790593901, 7.40066314358144070447698494444, 8.156776127075500786389965093130
