L(s) = 1 | − 2.22·2-s − 3.23·3-s + 2.93·4-s − 1.33·5-s + 7.19·6-s − 0.932·7-s − 2.06·8-s + 7.48·9-s + 2.95·10-s + 11-s − 9.49·12-s + 4.80·13-s + 2.07·14-s + 4.31·15-s − 1.27·16-s + 17-s − 16.6·18-s + 0.807·19-s − 3.90·20-s + 3.01·21-s − 2.22·22-s − 8.62·23-s + 6.69·24-s − 3.22·25-s − 10.6·26-s − 14.5·27-s − 2.73·28-s + ⋯ |
L(s) = 1 | − 1.57·2-s − 1.86·3-s + 1.46·4-s − 0.595·5-s + 2.93·6-s − 0.352·7-s − 0.730·8-s + 2.49·9-s + 0.934·10-s + 0.301·11-s − 2.74·12-s + 1.33·13-s + 0.553·14-s + 1.11·15-s − 0.317·16-s + 0.242·17-s − 3.92·18-s + 0.185·19-s − 0.872·20-s + 0.658·21-s − 0.473·22-s − 1.79·23-s + 1.36·24-s − 0.645·25-s − 2.09·26-s − 2.79·27-s − 0.516·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2804945036\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2804945036\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 1.33T + 5T^{2} \) |
| 7 | \( 1 + 0.932T + 7T^{2} \) |
| 13 | \( 1 - 4.80T + 13T^{2} \) |
| 19 | \( 1 - 0.807T + 19T^{2} \) |
| 23 | \( 1 + 8.62T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 - 8.01T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.149T + 61T^{2} \) |
| 67 | \( 1 + 9.07T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 8.57T + 83T^{2} \) |
| 89 | \( 1 + 8.59T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−7.83845281005894374469854676882, −7.15866807833945171617746341290, −6.54894257360345504396032386642, −5.96375201992037900879277274171, −5.39430423807893619429265521672, −4.13042621092407835923810269827, −3.83011588483539691511688229202, −2.08941140610068219895152253812, −1.17798207000594203089690592035, −0.44109729313636911525899981725,
0.44109729313636911525899981725, 1.17798207000594203089690592035, 2.08941140610068219895152253812, 3.83011588483539691511688229202, 4.13042621092407835923810269827, 5.39430423807893619429265521672, 5.96375201992037900879277274171, 6.54894257360345504396032386642, 7.15866807833945171617746341290, 7.83845281005894374469854676882