L(s) = 1 | − 0.311·2-s + 3-s − 1.90·4-s − 0.311·6-s − 2.68·7-s + 1.21·8-s + 9-s − 0.214·11-s − 1.90·12-s + 3.11·13-s + 0.836·14-s + 3.42·16-s − 0.474·17-s − 0.311·18-s − 19-s − 2.68·21-s + 0.0666·22-s − 2.47·23-s + 1.21·24-s − 0.969·26-s + 27-s + 5.11·28-s − 2.83·29-s − 4.90·31-s − 3.49·32-s − 0.214·33-s + 0.147·34-s + ⋯ |
L(s) = 1 | − 0.219·2-s + 0.577·3-s − 0.951·4-s − 0.127·6-s − 1.01·7-s + 0.429·8-s + 0.333·9-s − 0.0646·11-s − 0.549·12-s + 0.864·13-s + 0.223·14-s + 0.857·16-s − 0.115·17-s − 0.0733·18-s − 0.229·19-s − 0.586·21-s + 0.0142·22-s − 0.515·23-s + 0.247·24-s − 0.190·26-s + 0.192·27-s + 0.967·28-s − 0.526·29-s − 0.880·31-s − 0.617·32-s − 0.0373·33-s + 0.0253·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 + 0.474T + 17T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 + 4.90T + 31T^{2} \) |
| 37 | \( 1 + 1.93T + 37T^{2} \) |
| 41 | \( 1 - 5.45T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 9.52T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 6.56T + 59T^{2} \) |
| 61 | \( 1 - 7.76T + 61T^{2} \) |
| 67 | \( 1 - 6.85T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−9.198829885586550735868795207196, −8.440681092430085715008167916683, −7.78367599426033531317510983669, −6.71211207356515270365347783184, −5.88797657749043086184553529308, −4.80361172845398240614683529671, −3.78834785575388287206844104867, −3.20177434031997594542115518517, −1.64908103950759316887867763191, 0,
1.64908103950759316887867763191, 3.20177434031997594542115518517, 3.78834785575388287206844104867, 4.80361172845398240614683529671, 5.88797657749043086184553529308, 6.71211207356515270365347783184, 7.78367599426033531317510983669, 8.440681092430085715008167916683, 9.198829885586550735868795207196