L(s) = 1 | − 2-s + 3.00·3-s + 4-s − 1.36·5-s − 3.00·6-s − 0.145·7-s − 8-s + 6.05·9-s + 1.36·10-s − 0.231·11-s + 3.00·12-s − 5.52·13-s + 0.145·14-s − 4.10·15-s + 16-s − 1.01·17-s − 6.05·18-s − 0.303·19-s − 1.36·20-s − 0.439·21-s + 0.231·22-s − 0.101·23-s − 3.00·24-s − 3.13·25-s + 5.52·26-s + 9.18·27-s − 0.145·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 0.5·4-s − 0.610·5-s − 1.22·6-s − 0.0551·7-s − 0.353·8-s + 2.01·9-s + 0.431·10-s − 0.0697·11-s + 0.868·12-s − 1.53·13-s + 0.0390·14-s − 1.05·15-s + 0.250·16-s − 0.247·17-s − 1.42·18-s − 0.0697·19-s − 0.305·20-s − 0.0958·21-s + 0.0492·22-s − 0.0212·23-s − 0.614·24-s − 0.627·25-s + 1.08·26-s + 1.76·27-s − 0.0275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 - 3.00T + 3T^{2} \) |
| 5 | \( 1 + 1.36T + 5T^{2} \) |
| 7 | \( 1 + 0.145T + 7T^{2} \) |
| 11 | \( 1 + 0.231T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 + 0.303T + 19T^{2} \) |
| 23 | \( 1 + 0.101T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 9.57T + 47T^{2} \) |
| 53 | \( 1 + 5.01T + 53T^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 + 7.45T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 8.51T + 73T^{2} \) |
| 79 | \( 1 - 0.808T + 79T^{2} \) |
| 83 | \( 1 - 6.73T + 83T^{2} \) |
| 89 | \( 1 - 8.26T + 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−7.913513253976849362527613644636, −7.64665795284303576168582630950, −7.18877358968015519350917957764, −6.07566271527846939209590173444, −4.81937931492236260888119084807, −4.00450763477335884356761103494, −3.20362049671980802946256228192, −2.43253398034505313610463023314, −1.70598750563033730660058913289, 0,
1.70598750563033730660058913289, 2.43253398034505313610463023314, 3.20362049671980802946256228192, 4.00450763477335884356761103494, 4.81937931492236260888119084807, 6.07566271527846939209590173444, 7.18877358968015519350917957764, 7.64665795284303576168582630950, 7.913513253976849362527613644636