| L(s) = 1 | − 1.40·2-s − 3.19·3-s − 0.0217·4-s − 0.295·5-s + 4.49·6-s + 2.84·8-s + 7.20·9-s + 0.415·10-s − 0.372·11-s + 0.0694·12-s − 3.46·13-s + 0.944·15-s − 3.95·16-s − 3.29·17-s − 10.1·18-s − 19-s + 0.00643·20-s + 0.524·22-s − 7.41·23-s − 9.08·24-s − 4.91·25-s + 4.87·26-s − 13.4·27-s + 3.21·29-s − 1.32·30-s + 3.96·31-s − 0.123·32-s + ⋯ |
| L(s) = 1 | − 0.994·2-s − 1.84·3-s − 0.0108·4-s − 0.132·5-s + 1.83·6-s + 1.00·8-s + 2.40·9-s + 0.131·10-s − 0.112·11-s + 0.0200·12-s − 0.962·13-s + 0.243·15-s − 0.989·16-s − 0.799·17-s − 2.38·18-s − 0.229·19-s + 0.00143·20-s + 0.111·22-s − 1.54·23-s − 1.85·24-s − 0.982·25-s + 0.956·26-s − 2.58·27-s + 0.597·29-s − 0.242·30-s + 0.712·31-s − 0.0217·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2245186166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2245186166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.40T + 2T^{2} \) |
| 3 | \( 1 + 3.19T + 3T^{2} \) |
| 5 | \( 1 + 0.295T + 5T^{2} \) |
| 11 | \( 1 + 0.372T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 + 6.08T + 43T^{2} \) |
| 47 | \( 1 + 6.82T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 4.68T + 59T^{2} \) |
| 61 | \( 1 + 7.09T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 - 1.88T + 71T^{2} \) |
| 73 | \( 1 - 1.01T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 - 0.235T + 97T^{2} \) |
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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
−10.04937744778042793990583330902, −9.634918809832754639160964395878, −8.324030988194300629352162991660, −7.53596606519734735261624416400, −6.67137462746855371338861932999, −5.85309350132070740248440589553, −4.77471403617911288026437956823, −4.24220502466199952309004201093, −1.94082089896720299647658915095, −0.46960085043576933470813815231,
0.46960085043576933470813815231, 1.94082089896720299647658915095, 4.24220502466199952309004201093, 4.77471403617911288026437956823, 5.85309350132070740248440589553, 6.67137462746855371338861932999, 7.53596606519734735261624416400, 8.324030988194300629352162991660, 9.634918809832754639160964395878, 10.04937744778042793990583330902
