| L(s) = 1 | − 2.37i·2-s − 3.66·4-s + (−0.479 + 2.18i)5-s + 2.28i·7-s + 3.95i·8-s + (5.19 + 1.14i)10-s − 1.12·11-s − 5.95i·13-s + 5.43·14-s + 2.09·16-s − 5.80i·17-s + 4.08·19-s + (1.75 − 8.00i)20-s + 2.67i·22-s − i·23-s + ⋯ |
| L(s) = 1 | − 1.68i·2-s − 1.83·4-s + (−0.214 + 0.976i)5-s + 0.863i·7-s + 1.39i·8-s + (1.64 + 0.361i)10-s − 0.338·11-s − 1.65i·13-s + 1.45·14-s + 0.523·16-s − 1.40i·17-s + 0.936·19-s + (0.393 − 1.78i)20-s + 0.570i·22-s − 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8848792883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8848792883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.479 - 2.18i)T \) |
| 23 | \( 1 + iT \) |
| good | 2 | \( 1 + 2.37iT - 2T^{2} \) |
| 7 | \( 1 - 2.28iT - 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 + 5.95iT - 13T^{2} \) |
| 17 | \( 1 + 5.80iT - 17T^{2} \) |
| 19 | \( 1 - 4.08T + 19T^{2} \) |
| 29 | \( 1 + 0.408T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 9.80iT - 37T^{2} \) |
| 41 | \( 1 + 6.27T + 41T^{2} \) |
| 43 | \( 1 + 7.75iT - 43T^{2} \) |
| 47 | \( 1 - 6.40iT - 47T^{2} \) |
| 53 | \( 1 + 6.73iT - 53T^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 0.283iT - 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 9.61iT - 73T^{2} \) |
| 79 | \( 1 + 4.48T + 79T^{2} \) |
| 83 | \( 1 + 10.8iT - 83T^{2} \) |
| 89 | \( 1 - 5.68T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−9.718535169135213872426968032607, −9.077771675395509800649524740852, −7.973010809581692054613921730385, −7.14953692243781115147084942232, −5.71766490451868862229890331372, −4.99941452803444355977995942442, −3.54171733207713912146659262449, −2.95782078284939421929109940828, −2.20588603708131813236000865803, −0.41731855467763697973017240625,
1.45920096742545616899105700241, 3.80994932912505844787301880578, 4.50994509318557085187678786139, 5.29418149790115400169438849422, 6.28560590790786714862725535136, 7.02822075285825543528488304112, 7.82786858606984234264537654899, 8.452502032070016011044762773885, 9.233717727425825028791813722667, 9.960484505796030029892940064513
