L(s) = 1 | − 0.515·2-s − 0.702·3-s − 1.73·4-s − 5-s + 0.362·6-s + 0.362·7-s + 1.92·8-s − 2.50·9-s + 0.515·10-s − 1.40·11-s + 1.21·12-s + 1.51·13-s − 0.186·14-s + 0.702·15-s + 2.47·16-s − 3.17·17-s + 1.29·18-s + 5.87·19-s + 1.73·20-s − 0.254·21-s + 0.724·22-s − 4.36·23-s − 1.35·24-s + 25-s − 0.781·26-s + 3.86·27-s − 0.628·28-s + ⋯ |
L(s) = 1 | − 0.364·2-s − 0.405·3-s − 0.867·4-s − 0.447·5-s + 0.147·6-s + 0.136·7-s + 0.680·8-s − 0.835·9-s + 0.163·10-s − 0.423·11-s + 0.351·12-s + 0.420·13-s − 0.0499·14-s + 0.181·15-s + 0.618·16-s − 0.770·17-s + 0.304·18-s + 1.34·19-s + 0.387·20-s − 0.0555·21-s + 0.154·22-s − 0.909·23-s − 0.276·24-s + 0.200·25-s − 0.153·26-s + 0.744·27-s − 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 1201 | \( 1 + T \) |
good | 2 | \( 1 + 0.515T + 2T^{2} \) |
| 3 | \( 1 + 0.702T + 3T^{2} \) |
| 7 | \( 1 - 0.362T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 - 1.51T + 13T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 + 4.80T + 29T^{2} \) |
| 31 | \( 1 + 5.39T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 5.50T + 41T^{2} \) |
| 43 | \( 1 + 9.50T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 - 9.36T + 59T^{2} \) |
| 61 | \( 1 + 1.64T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 7.18T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 0.0314T + 97T^{2} \) |
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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
−7.86755656605471846254730271790, −7.23946953353645438820577697429, −6.16662953313755765709253635729, −5.50990075050402792244745709143, −4.93227992408313693792072033020, −4.05097691525498316465577465854, −3.39014601111554562719602802196, −2.25910247219054855445943506436, −0.946615029968896725193790151305, 0,
0.946615029968896725193790151305, 2.25910247219054855445943506436, 3.39014601111554562719602802196, 4.05097691525498316465577465854, 4.93227992408313693792072033020, 5.50990075050402792244745709143, 6.16662953313755765709253635729, 7.23946953353645438820577697429, 7.86755656605471846254730271790