L(s) = 1 | − 2.39·2-s − 3-s + 3.74·4-s + 1.84·5-s + 2.39·6-s − 7-s − 4.18·8-s + 9-s − 4.42·10-s + 4.68·11-s − 3.74·12-s + 6.22·13-s + 2.39·14-s − 1.84·15-s + 2.53·16-s + 3.83·17-s − 2.39·18-s + 2.42·19-s + 6.91·20-s + 21-s − 11.2·22-s − 3.97·23-s + 4.18·24-s − 1.59·25-s − 14.9·26-s − 27-s − 3.74·28-s + ⋯ |
L(s) = 1 | − 1.69·2-s − 0.577·3-s + 1.87·4-s + 0.825·5-s + 0.978·6-s − 0.377·7-s − 1.47·8-s + 0.333·9-s − 1.39·10-s + 1.41·11-s − 1.08·12-s + 1.72·13-s + 0.640·14-s − 0.476·15-s + 0.634·16-s + 0.930·17-s − 0.564·18-s + 0.555·19-s + 1.54·20-s + 0.218·21-s − 2.39·22-s − 0.828·23-s + 0.854·24-s − 0.318·25-s − 2.92·26-s − 0.192·27-s − 0.707·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.103623287\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.103623287\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 - 6.22T + 13T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 + 0.853T + 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 + 3.66T + 43T^{2} \) |
| 47 | \( 1 - 2.10T + 47T^{2} \) |
| 53 | \( 1 - 3.06T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 - 6.51T + 79T^{2} \) |
| 83 | \( 1 - 8.35T + 83T^{2} \) |
| 89 | \( 1 - 0.514T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.970541687667148555682144216045, −7.19079309172685448676436272677, −6.48797759446668658873587856796, −6.04395603816098890587968553334, −5.51865189455613311806676065936, −4.07678874878575387580225516299, −3.43910973003303506908845831705, −2.12663878228135405223570202526, −1.38257007693377304666123020810, −0.794628693174062463147245437995,
0.794628693174062463147245437995, 1.38257007693377304666123020810, 2.12663878228135405223570202526, 3.43910973003303506908845831705, 4.07678874878575387580225516299, 5.51865189455613311806676065936, 6.04395603816098890587968553334, 6.48797759446668658873587856796, 7.19079309172685448676436272677, 7.970541687667148555682144216045