| L(s) = 1 | − 2-s − 2.75·3-s + 4-s + 3.02·5-s + 2.75·6-s + 4.50·7-s − 8-s + 4.57·9-s − 3.02·10-s + 2.88·11-s − 2.75·12-s + 5.83·13-s − 4.50·14-s − 8.33·15-s + 16-s − 2.94·17-s − 4.57·18-s + 3.68·19-s + 3.02·20-s − 12.3·21-s − 2.88·22-s + 4.87·23-s + 2.75·24-s + 4.17·25-s − 5.83·26-s − 4.34·27-s + 4.50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.58·3-s + 0.5·4-s + 1.35·5-s + 1.12·6-s + 1.70·7-s − 0.353·8-s + 1.52·9-s − 0.957·10-s + 0.870·11-s − 0.794·12-s + 1.61·13-s − 1.20·14-s − 2.15·15-s + 0.250·16-s − 0.713·17-s − 1.07·18-s + 0.845·19-s + 0.677·20-s − 2.70·21-s − 0.615·22-s + 1.01·23-s + 0.561·24-s + 0.834·25-s − 1.14·26-s − 0.837·27-s + 0.850·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.820697273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.820697273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 - 3.02T + 5T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + 9.89T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 + 0.569T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 1.91T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 5.38T + 71T^{2} \) |
| 73 | \( 1 + 1.65T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.80245101032970496695899980113, −6.84787140154346808163992559986, −6.51977702530961771847837668442, −5.65264040263456969893627301209, −5.37255524539670868376586115384, −4.62147206738328677040557137722, −3.60449362214720498796623892990, −2.07333930693497711836382436101, −1.43125623250494435585041642635, −0.956029114172291353464680534794,
0.956029114172291353464680534794, 1.43125623250494435585041642635, 2.07333930693497711836382436101, 3.60449362214720498796623892990, 4.62147206738328677040557137722, 5.37255524539670868376586115384, 5.65264040263456969893627301209, 6.51977702530961771847837668442, 6.84787140154346808163992559986, 7.80245101032970496695899980113
