| L(s) = 1 | + (0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (3.44 + 0.733i)5-s + (−0.309 − 0.951i)6-s + (−2.03 + 1.68i)7-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (1.76 + 3.05i)10-s + (2.82 − 1.74i)11-s + (0.5 − 0.866i)12-s + (−1.54 + 4.74i)13-s + (−2.61 − 0.388i)14-s + (−2.85 − 2.07i)15-s + (−0.978 − 0.207i)16-s + (2.15 − 2.39i)17-s + ⋯ |
| L(s) = 1 | + (0.473 + 0.525i)2-s + (−0.527 − 0.234i)3-s + (−0.0522 + 0.497i)4-s + (1.54 + 0.327i)5-s + (−0.126 − 0.388i)6-s + (−0.771 + 0.636i)7-s + (−0.286 + 0.207i)8-s + (0.223 + 0.247i)9-s + (0.557 + 0.965i)10-s + (0.850 − 0.525i)11-s + (0.144 − 0.249i)12-s + (−0.427 + 1.31i)13-s + (−0.699 − 0.103i)14-s + (−0.736 − 0.535i)15-s + (−0.244 − 0.0519i)16-s + (0.522 − 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40503 + 1.12760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40503 + 1.12760i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 7 | \( 1 + (2.03 - 1.68i)T \) |
| 11 | \( 1 + (-2.82 + 1.74i)T \) |
| good | 5 | \( 1 + (-3.44 - 0.733i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (1.54 - 4.74i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.15 + 2.39i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.243 + 2.31i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (4.26 - 7.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.70 - 2.69i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-9.08 + 1.93i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (0.205 - 0.0914i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (3.16 - 2.29i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 9.81T + 43T^{2} \) |
| 47 | \( 1 + (0.984 + 9.36i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-3.05 + 0.649i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.0749 + 0.712i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-6.10 - 1.29i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (4.70 + 8.15i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.645 - 1.98i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.942 + 8.97i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (4.39 + 4.88i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (2.33 + 7.19i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.65 + 11.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.16 + 6.66i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.77189417731662571438788329934, −10.06588370992137528565746909924, −9.561730744504632391109547190147, −8.662801131109925604464545545401, −7.03711420789620200217095465835, −6.43434100490150115136320215357, −5.83405459097506352600688512617, −4.86281257526604348513595753737, −3.23980742909691685064445458281, −1.91305040924439528057577150435,
1.13625194550170279447214819454, 2.65732979926440816249227013145, 4.06183757194331196199940780753, 5.11937468255758018787674086126, 6.12396879789554436402698074594, 6.60033383317943584218751873805, 8.295090435267551323000688521887, 9.633555666797102983580772357847, 10.20382892399117999114884869626, 10.38604139183042947619181933810
