L(s) = 1 | + (0.707 + 0.707i)2-s + (−1 − 1.41i)3-s − 0.999i·4-s + (−1.41 − 1.41i)5-s + (0.292 − 1.70i)6-s + (−1 − i)7-s + (2.12 − 2.12i)8-s + (−1.00 + 2.82i)9-s − 2.00i·10-s + (−2.82 + 2.82i)11-s + (−1.41 + 0.999i)12-s − 1.41i·14-s + (−0.585 + 3.41i)15-s + 1.00·16-s + (−2.70 + 1.29i)18-s + (−1 + i)19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.577 − 0.816i)3-s − 0.499i·4-s + (−0.632 − 0.632i)5-s + (0.119 − 0.696i)6-s + (−0.377 − 0.377i)7-s + (0.750 − 0.750i)8-s + (−0.333 + 0.942i)9-s − 0.632i·10-s + (−0.852 + 0.852i)11-s + (−0.408 + 0.288i)12-s − 0.377i·14-s + (−0.151 + 0.881i)15-s + 0.250·16-s + (−0.638 + 0.304i)18-s + (−0.229 + 0.229i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.108694 - 0.670383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.108694 - 0.670383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1 + 1.41i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T + 2iT^{2} \) |
| 5 | \( 1 + (1.41 + 1.41i)T + 5iT^{2} \) |
| 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.82 - 2.82i)T - 11iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (1 - i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + (-5 + 5i)T - 31iT^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.41 + 1.41i)T + 41iT^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + (2.82 - 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (2.82 - 2.82i)T - 59iT^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-5 + 5i)T - 67iT^{2} \) |
| 71 | \( 1 + (-2.82 - 2.82i)T + 71iT^{2} \) |
| 73 | \( 1 + (1 + i)T + 73iT^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + (5.65 + 5.65i)T + 83iT^{2} \) |
| 89 | \( 1 + (-9.89 + 9.89i)T - 89iT^{2} \) |
| 97 | \( 1 + (7 - 7i)T - 97iT^{2} \) |
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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
−10.42732504324771031940393463089, −9.925198743538556442399299197626, −8.327790552389090681795235655832, −7.59181066812645919498436541365, −6.75899751472245250071524594636, −5.85971717573553252982609541817, −4.94803908289710508373848657373, −4.09101466394133302183611782626, −1.99684177936964598146125445298, −0.35295455859912775921321378120,
2.73324232175376469880236674273, 3.49778355333802456078703988908, 4.42871004452897072754881790645, 5.53266345040913408084528041862, 6.53910000091238888954378893823, 7.83822947520880574577364654393, 8.570875686313471260713322572244, 9.873635022493393608278915727678, 10.67737430389461016997130316578, 11.37133306044855106968724093528