| L(s) = 1 | + (0.422 + 1.98i)2-s + (−0.443 + 1.36i)3-s + (−1.95 + 0.869i)4-s + (0.366 − 3.48i)5-s + (−2.90 − 0.305i)6-s + (−1.27 − 1.14i)7-s + (−0.163 − 0.224i)8-s + (0.760 + 0.552i)9-s + (7.09 − 0.745i)10-s + 3.65i·11-s + (−0.320 − 3.05i)12-s + (−2.47 − 4.27i)13-s + (1.73 − 3.01i)14-s + (4.60 + 2.04i)15-s + (−2.48 + 2.75i)16-s + (−0.480 − 1.07i)17-s + ⋯ |
| L(s) = 1 | + (0.299 + 1.40i)2-s + (−0.256 + 0.788i)3-s + (−0.975 + 0.434i)4-s + (0.164 − 1.56i)5-s + (−1.18 − 0.124i)6-s + (−0.480 − 0.432i)7-s + (−0.0577 − 0.0794i)8-s + (0.253 + 0.184i)9-s + (2.24 − 0.235i)10-s + 1.10i·11-s + (−0.0925 − 0.880i)12-s + (−0.685 − 1.18i)13-s + (0.464 − 0.804i)14-s + (1.18 + 0.528i)15-s + (−0.620 + 0.688i)16-s + (−0.116 − 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.591542 + 0.727191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.591542 + 0.727191i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (6.86 + 3.72i)T \) |
| good | 2 | \( 1 + (-0.422 - 1.98i)T + (-1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (0.443 - 1.36i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.366 + 3.48i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (1.27 + 1.14i)T + (0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 - 3.65iT - 11T^{2} \) |
| 13 | \( 1 + (2.47 + 4.27i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.480 + 1.07i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (2.26 + 2.51i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.77 + 5.19i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.56 - 3.21i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.909 - 4.27i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (4.47 - 1.45i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.43 - 10.5i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (0.288 - 0.648i)T + (-28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (2.31 - 4.00i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.29 + 3.15i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.209 + 0.987i)T + (-53.8 + 23.9i)T^{2} \) |
| 67 | \( 1 + (-15.5 - 1.63i)T + (65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-0.515 + 0.0541i)T + (69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (0.0619 + 0.589i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.375 + 0.843i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.34 + 0.712i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-12.2 - 3.97i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-14.3 - 3.04i)T + (88.6 + 39.4i)T^{2} \) |
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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
−15.63669752452777084370990765537, −14.68906818396174685489510628102, −13.14811687375770381380402488668, −12.60942789320367606978979593038, −10.50606340563836616720668276161, −9.388569602878484937992045951070, −8.133029791413246294447327163454, −6.80448241295415933441457284965, −4.98194927886638326044415529501, −4.73025421501015838242517788734,
2.25412739760912315414010122288, 3.63577279204969512172053516409, 6.21356320138523719429777707743, 7.20045069472226624244628820940, 9.411451204383220496873981309659, 10.54460995668947796744953717684, 11.48456765180408986639684779342, 12.27094031286102647055075913578, 13.44599888256436960379978512867, 14.25048120821099812291388922128
