| L(s) = 1 | − 2.20·2-s + 2.85·4-s + 5-s + 4.87·7-s − 1.87·8-s − 2.20·10-s + 5.33·11-s + 2.01·13-s − 10.7·14-s − 1.57·16-s − 7.54·17-s + 3.88·19-s + 2.85·20-s − 11.7·22-s + 5.37·23-s + 25-s − 4.43·26-s + 13.8·28-s + 0.337·29-s − 7.70·31-s + 7.21·32-s + 16.6·34-s + 4.87·35-s + 3.90·37-s − 8.55·38-s − 1.87·40-s − 6.47·41-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 1.42·4-s + 0.447·5-s + 1.84·7-s − 0.662·8-s − 0.696·10-s + 1.60·11-s + 0.558·13-s − 2.86·14-s − 0.393·16-s − 1.83·17-s + 0.890·19-s + 0.637·20-s − 2.50·22-s + 1.12·23-s + 0.200·25-s − 0.870·26-s + 2.62·28-s + 0.0627·29-s − 1.38·31-s + 1.27·32-s + 2.85·34-s + 0.823·35-s + 0.642·37-s − 1.38·38-s − 0.296·40-s − 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.462406165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462406165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 13 | \( 1 - 2.01T + 13T^{2} \) |
| 17 | \( 1 + 7.54T + 17T^{2} \) |
| 19 | \( 1 - 3.88T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 - 0.337T + 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 - 3.90T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 1.00T + 43T^{2} \) |
| 47 | \( 1 - 7.38T + 47T^{2} \) |
| 53 | \( 1 - 7.81T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 61 | \( 1 - 9.00T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 - 4.54T + 73T^{2} \) |
| 79 | \( 1 - 0.958T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 97 | \( 1 - 0.264T + 97T^{2} \) |
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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
−8.757242383274577889653038056829, −7.928707074206884171822413764483, −7.07680684837122766962597027442, −6.69778088746490570517632140978, −5.56653236639636288034290666528, −4.69232616640360262832861156777, −3.90765500848259503590170918312, −2.34476303561727810741329083228, −1.60037180252512750458374335993, −0.996881450717174190400804119596,
0.996881450717174190400804119596, 1.60037180252512750458374335993, 2.34476303561727810741329083228, 3.90765500848259503590170918312, 4.69232616640360262832861156777, 5.56653236639636288034290666528, 6.69778088746490570517632140978, 7.07680684837122766962597027442, 7.928707074206884171822413764483, 8.757242383274577889653038056829
