| L(s) = 1 | + (1.04 − 0.957i)2-s + (−0.167 − 1.11i)3-s + (0.165 − 1.99i)4-s + (2.65 + 1.04i)5-s + (−1.23 − 0.995i)6-s + (2.56 + 0.648i)7-s + (−1.73 − 2.23i)8-s + (1.66 − 0.512i)9-s + (3.75 − 1.45i)10-s + (−1.29 + 4.20i)11-s + (−2.24 + 0.150i)12-s + (−1.42 + 0.324i)13-s + (3.29 − 1.78i)14-s + (0.711 − 3.11i)15-s + (−3.94 − 0.658i)16-s + (−0.726 + 0.495i)17-s + ⋯ |
| L(s) = 1 | + (0.735 − 0.677i)2-s + (−0.0966 − 0.641i)3-s + (0.0826 − 0.996i)4-s + (1.18 + 0.465i)5-s + (−0.505 − 0.406i)6-s + (0.969 + 0.245i)7-s + (−0.614 − 0.789i)8-s + (0.553 − 0.170i)9-s + (1.18 − 0.460i)10-s + (−0.390 + 1.26i)11-s + (−0.646 + 0.0433i)12-s + (−0.394 + 0.0900i)13-s + (0.879 − 0.476i)14-s + (0.183 − 0.804i)15-s + (−0.986 − 0.164i)16-s + (−0.176 + 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.86302 - 1.50991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86302 - 1.50991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.04 + 0.957i)T \) |
| 7 | \( 1 + (-2.56 - 0.648i)T \) |
| good | 3 | \( 1 + (0.167 + 1.11i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-2.65 - 1.04i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (1.29 - 4.20i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (1.42 - 0.324i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (0.726 - 0.495i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (4.43 - 2.55i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.97 + 3.39i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-2.61 + 5.42i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (3.42 - 5.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.38 - 0.253i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-2.18 - 2.74i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (6.60 + 5.26i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (1.71 - 1.59i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (10.7 + 0.808i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-10.0 + 3.94i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-8.05 + 0.603i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-11.5 - 6.66i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.46 + 0.703i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-11.2 - 10.4i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (5.25 + 9.11i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.53 - 0.806i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-5.35 + 1.65i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 9.24T + 97T^{2} \) |
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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
−11.19647822062277766343500405017, −10.05587751981127390315821902056, −9.933196904723043499606871764340, −8.320499638314972131317876793266, −6.98960982007924437417941156160, −6.27886731658997180159061124573, −5.17843682803193248857465203136, −4.21617370190446278153416152903, −2.19156901180089521942046125765, −1.87324056440732973272239883689,
2.09022796974832861510451386876, 3.78599105118565999125549991727, 4.93903306898490690990161792157, 5.42845550981580426938024729183, 6.52475803276192940514151692891, 7.78906710781726003287559279605, 8.639528333566361205855348903102, 9.611404662289265567371609535426, 10.70073825886027327554280662935, 11.42007122807183435190564014940
