| L(s) = 1 | + (−0.936 + 0.349i)2-s + (−1.69 + 0.926i)3-s + (0.755 − 0.654i)4-s + (−1.50 + 1.65i)5-s + (1.26 − 1.46i)6-s + (−0.735 − 0.983i)7-s + (−0.479 + 0.877i)8-s + (0.400 − 0.623i)9-s + (0.831 − 2.07i)10-s + (3.08 − 1.40i)11-s + (−0.675 + 1.81i)12-s + (−5.47 − 4.09i)13-s + (1.03 + 0.663i)14-s + (1.02 − 4.20i)15-s + (0.142 − 0.989i)16-s + (−0.214 − 3.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.662 + 0.247i)2-s + (−0.980 + 0.535i)3-s + (0.377 − 0.327i)4-s + (−0.672 + 0.739i)5-s + (0.517 − 0.596i)6-s + (−0.278 − 0.371i)7-s + (−0.169 + 0.310i)8-s + (0.133 − 0.207i)9-s + (0.262 − 0.656i)10-s + (0.929 − 0.424i)11-s + (−0.195 + 0.523i)12-s + (−1.51 − 1.13i)13-s + (0.276 + 0.177i)14-s + (0.263 − 1.08i)15-s + (0.0355 − 0.247i)16-s + (−0.0520 − 0.728i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0744477 - 0.101654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0744477 - 0.101654i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.936 - 0.349i)T \) |
| 5 | \( 1 + (1.50 - 1.65i)T \) |
| 23 | \( 1 + (-3.93 - 2.74i)T \) |
| good | 3 | \( 1 + (1.69 - 0.926i)T + (1.62 - 2.52i)T^{2} \) |
| 7 | \( 1 + (0.735 + 0.983i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (-3.08 + 1.40i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (5.47 + 4.09i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.214 + 3.00i)T + (-16.8 + 2.41i)T^{2} \) |
| 19 | \( 1 + (1.88 + 2.17i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (3.99 + 3.46i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (5.92 - 1.74i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (6.54 - 1.42i)T + (33.6 - 15.3i)T^{2} \) |
| 41 | \( 1 + (2.34 - 1.50i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-4.04 - 7.40i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (-5.79 - 5.79i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.41 - 6.30i)T + (14.9 - 50.8i)T^{2} \) |
| 59 | \( 1 + (7.66 - 1.10i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.704 - 2.39i)T + (-51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (3.20 + 8.58i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (1.74 - 3.83i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.769 - 0.0550i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (0.551 + 3.83i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (2.57 + 11.8i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (10.7 + 3.16i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-1.36 + 6.29i)T + (-88.2 - 40.2i)T^{2} \) |
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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
−11.51995840256672682263011156794, −10.93853650212412187859684638122, −10.13853201698311094473493244194, −9.190397499531943899489722547854, −7.69816118704523004714682296917, −6.99754068197144014184282874263, −5.83285344791873589559109150869, −4.64287029170280215612155229341, −3.04570539994886616519266297535, −0.14024909423369951732738841611,
1.74788057809745563927422739120, 3.96383858904517471312065310176, 5.32062150826245841746252128664, 6.69238278542788572220562377998, 7.34078258283906448478490157616, 8.818468854521609579754228131893, 9.377633839508352914422807762466, 10.78083544570333884100138972650, 11.70535447409036465962575144242, 12.41177443960686223081366337296
