L(s) = 1 | + 7.73i·2-s + (−8.52 + 2.87i)3-s − 43.8·4-s + 0.620i·5-s + (−22.2 − 65.9i)6-s + 71.8·7-s − 215. i·8-s + (64.4 − 49.0i)9-s − 4.79·10-s + 63.2i·11-s + (373. − 126. i)12-s + 204.·13-s + 555. i·14-s + (−1.78 − 5.28i)15-s + 963.·16-s − 443. i·17-s + ⋯ |
L(s) = 1 | + 1.93i·2-s + (−0.947 + 0.319i)3-s − 2.73·4-s + 0.0248i·5-s + (−0.617 − 1.83i)6-s + 1.46·7-s − 3.36i·8-s + (0.795 − 0.605i)9-s − 0.0479·10-s + 0.522i·11-s + (2.59 − 0.875i)12-s + 1.21·13-s + 2.83i·14-s + (−0.00792 − 0.0235i)15-s + 3.76·16-s − 1.53i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.258837812\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.258837812\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (8.52 - 2.87i)T \) |
| 59 | \( 1 - 453. iT \) |
good | 2 | \( 1 - 7.73iT - 16T^{2} \) |
| 5 | \( 1 - 0.620iT - 625T^{2} \) |
| 7 | \( 1 - 71.8T + 2.40e3T^{2} \) |
| 11 | \( 1 - 63.2iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 204.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 443. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 526.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 966. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 559. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.27e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 362.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.02e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 487.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 34.7iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.64e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 2.51e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.37e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 884. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 7.90e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.00e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.56e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 2.01e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 7.87e3T + 8.85e7T^{2} \) |
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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
−12.53012643761737985981154416995, −11.24734398923528526586947953244, −10.15204732602088117367151521119, −8.831772955424117641271686093135, −8.080625937760840951893892370729, −6.84465062409007306687602050346, −6.12409771653237397790488313067, −4.71994593251977024007484785258, −4.54753443408235220506996248444, −0.72306622585568815157350798425,
1.09493221586094133502423362111, 1.83016729284211696518939224970, 3.78849827351260225455606092676, 4.80163003844617578844253360492, 5.92778341791775347221491323061, 8.097250838738205760081351267635, 8.787679909896149179826182566468, 10.45769022720473360243418254640, 10.88736462829261726523817694496, 11.48643093108180396513886159043