L(s) = 1 | + 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (4.91 + 0.894i)5-s − 2.44·6-s − 6.65·7-s − 2.82i·8-s − 2.99·9-s + (−1.26 + 6.95i)10-s + 10.9i·11-s − 3.46i·12-s − 2.68i·13-s − 9.41i·14-s + (−1.54 + 8.52i)15-s + 4.00·16-s − 26.7·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (0.983 + 0.178i)5-s − 0.408·6-s − 0.950·7-s − 0.353i·8-s − 0.333·9-s + (−0.126 + 0.695i)10-s + 0.998i·11-s − 0.288i·12-s − 0.206i·13-s − 0.672i·14-s + (−0.103 + 0.568i)15-s + 0.250·16-s − 1.57·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4557274754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4557274754\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + (-4.91 - 0.894i)T \) |
| 23 | \( 1 + (7.63 - 21.6i)T \) |
good | 7 | \( 1 + 6.65T + 49T^{2} \) |
| 11 | \( 1 - 10.9iT - 121T^{2} \) |
| 13 | \( 1 + 2.68iT - 169T^{2} \) |
| 17 | \( 1 + 26.7T + 289T^{2} \) |
| 19 | \( 1 + 21.8iT - 361T^{2} \) |
| 29 | \( 1 + 24.5T + 841T^{2} \) |
| 31 | \( 1 - 19.2T + 961T^{2} \) |
| 37 | \( 1 + 22.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 66.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 3.50iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 77.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 7.04T + 4.48e3T^{2} \) |
| 71 | \( 1 - 104.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 111. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 55.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 61.5T + 9.40e3T^{2} \) |
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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
−10.53283575619174479855335186379, −9.674134360344079155002292591023, −9.376809914387626528535501620471, −8.389320790537331370765524104494, −6.90503048300779149195726740507, −6.63460573194153670435709646558, −5.44170264405237905321432913829, −4.68440144228834957668500576836, −3.42140578524163240638231283171, −2.12795681489989203016532622571,
0.15030720922982821563473859958, 1.66764650154557165065665299146, 2.67705690702188136518554823831, 3.78729336392245915551972165864, 5.18684505088006279538941017562, 6.22344213309461966236129403078, 6.71046661639973136084382033117, 8.309389316617615817722144317605, 8.870590788050630814202687773981, 9.806343791998833994435709025791