| L(s) = 1 | − 2.22·2-s + 1.47·3-s + 2.96·4-s − 3.29·6-s + 4.42·7-s − 2.15·8-s − 0.812·9-s − 11-s + 4.38·12-s + 3.92·13-s − 9.86·14-s − 1.13·16-s + 1.61·17-s + 1.81·18-s + 19-s + 6.54·21-s + 2.22·22-s − 0.113·23-s − 3.18·24-s − 8.75·26-s − 5.63·27-s + 13.1·28-s + 1.08·29-s − 4.17·31-s + 6.83·32-s − 1.47·33-s − 3.59·34-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.853·3-s + 1.48·4-s − 1.34·6-s + 1.67·7-s − 0.762·8-s − 0.270·9-s − 0.301·11-s + 1.26·12-s + 1.08·13-s − 2.63·14-s − 0.282·16-s + 0.391·17-s + 0.427·18-s + 0.229·19-s + 1.42·21-s + 0.475·22-s − 0.0236·23-s − 0.650·24-s − 1.71·26-s − 1.08·27-s + 2.48·28-s + 0.201·29-s − 0.749·31-s + 1.20·32-s − 0.257·33-s − 0.617·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.563249248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.563249248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 3 | \( 1 - 1.47T + 3T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 23 | \( 1 + 0.113T + 23T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 0.883T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 4.28T + 71T^{2} \) |
| 73 | \( 1 - 1.92T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 - 3.33T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.336590790983827096246798319118, −7.78327708317457051889673808428, −7.37155386237562960494025484193, −6.23611808900795718302292538494, −5.39706959678756601572935967398, −4.46476869028552449083531091576, −3.46542153791902390847598288143, −2.40782520568370742672461272781, −1.73295852480627462967350414203, −0.871098282183441681635004060858,
0.871098282183441681635004060858, 1.73295852480627462967350414203, 2.40782520568370742672461272781, 3.46542153791902390847598288143, 4.46476869028552449083531091576, 5.39706959678756601572935967398, 6.23611808900795718302292538494, 7.37155386237562960494025484193, 7.78327708317457051889673808428, 8.336590790983827096246798319118
