L(s) = 1 | + 8.95·2-s − 25.1·3-s + 48.2·4-s − 61.2·5-s − 224.·6-s − 166.·7-s + 145.·8-s + 387.·9-s − 549.·10-s + 607.·11-s − 1.21e3·12-s − 1.03e3·13-s − 1.48e3·14-s + 1.53e3·15-s − 238.·16-s + 1.43e3·17-s + 3.46e3·18-s − 1.33e3·19-s − 2.95e3·20-s + 4.16e3·21-s + 5.44e3·22-s − 437.·23-s − 3.65e3·24-s + 631.·25-s − 9.31e3·26-s − 3.62e3·27-s − 8.01e3·28-s + ⋯ |
L(s) = 1 | + 1.58·2-s − 1.61·3-s + 1.50·4-s − 1.09·5-s − 2.55·6-s − 1.28·7-s + 0.805·8-s + 1.59·9-s − 1.73·10-s + 1.51·11-s − 2.42·12-s − 1.70·13-s − 2.02·14-s + 1.76·15-s − 0.233·16-s + 1.20·17-s + 2.52·18-s − 0.846·19-s − 1.65·20-s + 2.06·21-s + 2.39·22-s − 0.172·23-s − 1.29·24-s + 0.202·25-s − 2.70·26-s − 0.956·27-s − 1.93·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + 1.84e3T \) |
good | 2 | \( 1 - 8.95T + 32T^{2} \) |
| 3 | \( 1 + 25.1T + 243T^{2} \) |
| 5 | \( 1 + 61.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 166.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 607.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.43e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 437.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 87.2T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.67e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.01e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 8.62e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.80e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.17e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.25e4T + 8.58e9T^{2} \) |
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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
−14.41282445733040318445797365360, −12.64555570128022482923004842779, −12.16104233849528165636181436436, −11.53210726899759418637940743177, −9.915471721630831233830866232692, −7.03253982674949730631962237087, −6.20499874402488362728708371144, −4.80581840398783684321356251254, −3.61057028552080163641041369726, 0,
3.61057028552080163641041369726, 4.80581840398783684321356251254, 6.20499874402488362728708371144, 7.03253982674949730631962237087, 9.915471721630831233830866232692, 11.53210726899759418637940743177, 12.16104233849528165636181436436, 12.64555570128022482923004842779, 14.41282445733040318445797365360