L(s) = 1 | + (−0.892 − 0.239i)2-s + (0.988 − 0.570i)3-s + (−0.993 − 0.573i)4-s + (−2.80 − 2.80i)5-s + (−1.01 + 0.272i)6-s + (2.12 − 1.57i)7-s + (2.05 + 2.05i)8-s + (−0.848 + 1.47i)9-s + (1.82 + 3.16i)10-s + (0.544 − 2.03i)11-s − 1.30·12-s + (3.41 − 1.16i)13-s + (−2.27 + 0.893i)14-s + (−4.36 − 1.16i)15-s + (−0.195 − 0.339i)16-s + (−1.58 + 2.74i)17-s + ⋯ |
L(s) = 1 | + (−0.630 − 0.169i)2-s + (0.570 − 0.329i)3-s + (−0.496 − 0.286i)4-s + (−1.25 − 1.25i)5-s + (−0.415 + 0.111i)6-s + (0.804 − 0.594i)7-s + (0.726 + 0.726i)8-s + (−0.282 + 0.490i)9-s + (0.578 + 1.00i)10-s + (0.164 − 0.613i)11-s − 0.377·12-s + (0.946 − 0.324i)13-s + (−0.607 + 0.238i)14-s + (−1.12 − 0.301i)15-s + (−0.0489 − 0.0847i)16-s + (−0.384 + 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.451196 - 0.502122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.451196 - 0.502122i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.12 + 1.57i)T \) |
| 13 | \( 1 + (-3.41 + 1.16i)T \) |
good | 2 | \( 1 + (0.892 + 0.239i)T + (1.73 + i)T^{2} \) |
| 3 | \( 1 + (-0.988 + 0.570i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.80 + 2.80i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.544 + 2.03i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (1.58 - 2.74i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 + 0.302i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.26 + 1.88i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.584 + 1.01i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.57 - 3.57i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.14 + 4.26i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.85 + 6.93i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.91 + 1.10i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (8.21 - 8.21i)T - 47iT^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 + (-0.0633 - 0.236i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-10.7 - 6.18i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.95 + 2.66i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.60 - 5.98i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (3.47 - 3.47i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 + (-3.31 - 3.31i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.71 - 1.80i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (15.8 - 4.24i)T + (84.0 - 48.5i)T^{2} \) |
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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
−13.69938874575319452330240128018, −12.94350697615185947179972845011, −11.42837040235937696487839974410, −10.69846929470157872985055912139, −8.809065611643694340022670280504, −8.454693113769158443902825150573, −7.66459889030041382708701934140, −5.19836774581948512780317372565, −4.02718190754186313894289292193, −1.12940559217859421192139152019,
3.21567724563409809160966855716, 4.39804818711276359345084503789, 6.77891265553642557501566193755, 7.908197013967134849294754436797, 8.684297712699076232643435670821, 9.788696978346026720041775543940, 11.22760725361810793900683895499, 11.89363475128794373792762122533, 13.57314012504888595994465291129, 14.72277160682044266792861977875