| L(s) = 1 | − 1.93·3-s − 5-s − 2.38·7-s + 0.735·9-s − 5.33·11-s − 4.53·13-s + 1.93·15-s + 1.81·17-s − 7.00·19-s + 4.60·21-s + 23-s + 25-s + 4.37·27-s − 0.118·29-s + 0.884·31-s + 10.3·33-s + 2.38·35-s + 7.51·37-s + 8.77·39-s − 1.45·41-s − 0.735·45-s − 10.4·47-s − 1.32·49-s − 3.50·51-s − 9.42·53-s + 5.33·55-s + 13.5·57-s + ⋯ |
| L(s) = 1 | − 1.11·3-s − 0.447·5-s − 0.900·7-s + 0.245·9-s − 1.60·11-s − 1.25·13-s + 0.499·15-s + 0.440·17-s − 1.60·19-s + 1.00·21-s + 0.208·23-s + 0.200·25-s + 0.842·27-s − 0.0219·29-s + 0.158·31-s + 1.79·33-s + 0.402·35-s + 1.23·37-s + 1.40·39-s − 0.226·41-s − 0.109·45-s − 1.52·47-s − 0.189·49-s − 0.491·51-s − 1.29·53-s + 0.719·55-s + 1.79·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2478813656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2478813656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 29 | \( 1 + 0.118T + 29T^{2} \) |
| 31 | \( 1 - 0.884T + 31T^{2} \) |
| 37 | \( 1 - 7.51T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 9.42T + 53T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 + 2.80T + 61T^{2} \) |
| 67 | \( 1 - 3.11T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 6.80T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 2.89T + 89T^{2} \) |
| 97 | \( 1 + 1.97T + 97T^{2} \) |
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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
−9.491403231926375014812525892123, −8.257065013165311266685521909733, −7.69066969321607573581739248795, −6.66916328489374909934527829284, −6.12595048408386719338227348832, −5.10963416088376515193759381939, −4.64863388003871234103167066276, −3.25875493881566695097223488837, −2.37603137320533751218488741797, −0.32957100908932285842405515070,
0.32957100908932285842405515070, 2.37603137320533751218488741797, 3.25875493881566695097223488837, 4.64863388003871234103167066276, 5.10963416088376515193759381939, 6.12595048408386719338227348832, 6.66916328489374909934527829284, 7.69066969321607573581739248795, 8.257065013165311266685521909733, 9.491403231926375014812525892123
