L(s) = 1 | + 2-s − 2.69·3-s + 4-s − 2.92·5-s − 2.69·6-s + 3.32·7-s + 8-s + 4.27·9-s − 2.92·10-s + 1.11·11-s − 2.69·12-s − 3.39·13-s + 3.32·14-s + 7.90·15-s + 16-s + 1.22·17-s + 4.27·18-s − 6.86·19-s − 2.92·20-s − 8.95·21-s + 1.11·22-s − 1.09·23-s − 2.69·24-s + 3.58·25-s − 3.39·26-s − 3.43·27-s + 3.32·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.55·3-s + 0.5·4-s − 1.31·5-s − 1.10·6-s + 1.25·7-s + 0.353·8-s + 1.42·9-s − 0.926·10-s + 0.335·11-s − 0.778·12-s − 0.942·13-s + 0.887·14-s + 2.04·15-s + 0.250·16-s + 0.298·17-s + 1.00·18-s − 1.57·19-s − 0.655·20-s − 1.95·21-s + 0.237·22-s − 0.228·23-s − 0.550·24-s + 0.716·25-s − 0.666·26-s − 0.660·27-s + 0.627·28-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 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 - 1.11T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 - 1.22T + 17T^{2} \) |
| 19 | \( 1 + 6.86T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 + 0.159T + 29T^{2} \) |
| 31 | \( 1 - 9.27T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 0.437T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 + 3.02T + 47T^{2} \) |
| 53 | \( 1 - 4.74T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 79 | \( 1 + 6.06T + 79T^{2} \) |
| 83 | \( 1 - 3.64T + 83T^{2} \) |
| 89 | \( 1 + 4.57T + 89T^{2} \) |
| 97 | \( 1 - 0.713T + 97T^{2} \) |
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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
−7.960752687954594133622513170013, −7.16630303985253123516292137623, −6.50509214056275484912379127453, −5.76353270398677537668162280280, −4.78051511199759868756574918165, −4.59722719919368067476354024587, −3.88081554969474226358371292436, −2.49052238031343481162580598288, −1.23348554139125732401695472871, 0,
1.23348554139125732401695472871, 2.49052238031343481162580598288, 3.88081554969474226358371292436, 4.59722719919368067476354024587, 4.78051511199759868756574918165, 5.76353270398677537668162280280, 6.50509214056275484912379127453, 7.16630303985253123516292137623, 7.960752687954594133622513170013