L(s) = 1 | + (3.53 + 4.41i)2-s + 21.4·3-s + (−7.01 + 31.2i)4-s + (48.3 − 27.9i)5-s + (75.7 + 94.6i)6-s − 39.9i·7-s + (−162. + 79.4i)8-s + 216.·9-s + (294. + 114. i)10-s − 141. i·11-s + (−150. + 669. i)12-s − 700.·13-s + (176. − 141. i)14-s + (1.03e3 − 600. i)15-s + (−925. − 437. i)16-s + 960. i·17-s + ⋯ |
L(s) = 1 | + (0.624 + 0.780i)2-s + 1.37·3-s + (−0.219 + 0.975i)4-s + (0.865 − 0.500i)5-s + (0.859 + 1.07i)6-s − 0.307i·7-s + (−0.898 + 0.438i)8-s + 0.890·9-s + (0.931 + 0.362i)10-s − 0.352i·11-s + (−0.301 + 1.34i)12-s − 1.14·13-s + (0.240 − 0.192i)14-s + (1.19 − 0.688i)15-s + (−0.904 − 0.427i)16-s + 0.805i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.90795 + 1.54846i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.90795 + 1.54846i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.53 - 4.41i)T \) |
| 5 | \( 1 + (-48.3 + 27.9i)T \) |
good | 3 | \( 1 - 21.4T + 243T^{2} \) |
| 7 | \( 1 + 39.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 141. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 700.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 960. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.20e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 4.49e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.83e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.12e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.15e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.73e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.51e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 7.11e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.17e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.03e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.37e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 881.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.05e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.22e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.55e5iT - 8.58e9T^{2} \) |
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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
−14.80881752229573298877447567648, −14.34711043238015098256825449211, −13.31897066403165843299749387588, −12.40424982591242346356671265219, −9.977283957993028748703701154802, −8.722761302233102441810306054547, −7.78072198053155134141558847780, −6.04701434477008679069945213910, −4.26725486175456754297750535377, −2.50560932291624867661723469332,
2.11837533100772073822472656714, 3.10576413838540253349091512602, 5.15225089610035559091191488181, 7.13202553510804183998440500301, 9.211648033482409402950259940060, 9.767976259701224483030325049690, 11.38466031445070954334600772760, 12.94593490329912918251262786402, 13.80012526101483180688364698732, 14.63243025229219939769963539626