| L(s) = 1 | + 2-s − 3.33·3-s + 4-s − 2.81·5-s − 3.33·6-s + 2.39·7-s + 8-s + 8.09·9-s − 2.81·10-s + 2.12·11-s − 3.33·12-s + 1.83·13-s + 2.39·14-s + 9.38·15-s + 16-s − 5.43·17-s + 8.09·18-s − 1.66·19-s − 2.81·20-s − 7.99·21-s + 2.12·22-s + 0.107·23-s − 3.33·24-s + 2.93·25-s + 1.83·26-s − 16.9·27-s + 2.39·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.92·3-s + 0.5·4-s − 1.25·5-s − 1.35·6-s + 0.906·7-s + 0.353·8-s + 2.69·9-s − 0.890·10-s + 0.639·11-s − 0.961·12-s + 0.507·13-s + 0.641·14-s + 2.42·15-s + 0.250·16-s − 1.31·17-s + 1.90·18-s − 0.381·19-s − 0.629·20-s − 1.74·21-s + 0.452·22-s + 0.0223·23-s − 0.679·24-s + 0.587·25-s + 0.359·26-s − 3.26·27-s + 0.453·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 4013 | \( 1+O(T) \) |
| good | 3 | \( 1 + 3.33T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 - 0.107T + 23T^{2} \) |
| 29 | \( 1 + 1.06T + 29T^{2} \) |
| 31 | \( 1 - 1.63T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 + 8.30T + 41T^{2} \) |
| 43 | \( 1 - 0.583T + 43T^{2} \) |
| 47 | \( 1 + 4.59T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 1.74T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 3.04T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 9.59T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 - 2.50T + 97T^{2} \) |
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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
−7.12726218154210255143797168000, −6.69332936058088454245414523658, −6.09174050752025048742689227950, −5.26364700412615247789933329929, −4.63755134026177887866789858685, −4.22829687550470672677541957291, −3.60472738156879226346683368680, −2.01098676753543280468412063288, −1.09881023533053178101913166755, 0,
1.09881023533053178101913166755, 2.01098676753543280468412063288, 3.60472738156879226346683368680, 4.22829687550470672677541957291, 4.63755134026177887866789858685, 5.26364700412615247789933329929, 6.09174050752025048742689227950, 6.69332936058088454245414523658, 7.12726218154210255143797168000
