| L(s) = 1 | + (−0.190 + 0.587i)2-s + (−0.669 + 0.743i)3-s + (1.30 + 0.951i)4-s + (1.30 − 2.26i)5-s + (−0.309 − 0.535i)6-s + (−0.313 + 2.98i)7-s + (−1.80 + 1.31i)8-s + (0.209 + 1.98i)9-s + (1.08 + 1.20i)10-s + (−0.697 + 0.310i)11-s + (−1.58 + 0.336i)12-s + (−4.74 − 1.00i)13-s + (−1.69 − 0.754i)14-s + (0.809 + 2.48i)15-s + (0.572 + 1.76i)16-s + (0.215 + 0.0960i)17-s + ⋯ |
| L(s) = 1 | + (−0.135 + 0.415i)2-s + (−0.386 + 0.429i)3-s + (0.654 + 0.475i)4-s + (0.585 − 1.01i)5-s + (−0.126 − 0.218i)6-s + (−0.118 + 1.12i)7-s + (−0.639 + 0.464i)8-s + (0.0696 + 0.663i)9-s + (0.342 + 0.380i)10-s + (−0.210 + 0.0936i)11-s + (−0.456 + 0.0971i)12-s + (−1.31 − 0.279i)13-s + (−0.452 − 0.201i)14-s + (0.208 + 0.642i)15-s + (0.143 + 0.440i)16-s + (0.0523 + 0.0232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.233194 + 1.14058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.233194 + 1.14058i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (0.190 - 0.587i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.669 - 0.743i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (-1.30 + 2.26i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.313 - 2.98i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (0.697 - 0.310i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (4.74 + 1.00i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.215 - 0.0960i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (4.89 - 1.03i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-4.42 + 3.21i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.66 - 8.19i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.118 + 0.204i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.33 - 4.80i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-4.51 + 0.960i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (1.04 + 3.21i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.32 - 12.6i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-6.33 + 7.03i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 + (-2.11 + 3.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.00942 + 0.0896i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (7.82 - 3.48i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (2.73 + 3.03i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-5.16 - 3.75i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.28 - 3.11i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.42388197095120562935667287665, −9.328106372894389225880127137983, −8.782947819206602599624125258235, −7.943846834798567279355762422803, −7.01878318117457280420401495458, −5.89624753236698461883425133701, −5.30863595834173283463074278809, −4.57314039926330855195887907847, −2.81214609046502376923362719476, −1.99563207359567482999859754532,
0.53757221368283849994529731969, 2.00147103659247028781458586740, 2.91417491809159066820847584683, 4.13750987003398233053581278421, 5.60072265705333859533539129919, 6.43657270001657871889072991277, 7.00440281939830079961468161384, 7.52537869672231565722411912074, 9.241542133357033260954041039697, 9.930086176723303761980926649601
