| L(s) = 1 | + (−0.786 − 0.618i)2-s + (−1.20 + 1.69i)3-s + (0.235 + 0.971i)4-s + (−0.434 + 0.0836i)5-s + (1.99 − 0.586i)6-s + (2.62 + 0.297i)7-s + (0.415 − 0.909i)8-s + (−0.438 − 1.26i)9-s + (0.393 + 0.202i)10-s + (1.43 − 1.13i)11-s + (−1.93 − 0.774i)12-s + (−5.14 + 3.30i)13-s + (−1.88 − 1.85i)14-s + (0.382 − 0.838i)15-s + (−0.888 + 0.458i)16-s + (2.17 + 2.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.437i)2-s + (−0.697 + 0.979i)3-s + (0.117 + 0.485i)4-s + (−0.194 + 0.0374i)5-s + (0.816 − 0.239i)6-s + (0.993 + 0.112i)7-s + (0.146 − 0.321i)8-s + (−0.146 − 0.422i)9-s + (0.124 + 0.0640i)10-s + (0.433 − 0.341i)11-s + (−0.558 − 0.223i)12-s + (−1.42 + 0.917i)13-s + (−0.503 − 0.496i)14-s + (0.0988 − 0.216i)15-s + (−0.222 + 0.114i)16-s + (0.528 + 0.503i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.398792 + 0.530800i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.398792 + 0.530800i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (-2.62 - 0.297i)T \) |
| 23 | \( 1 + (0.185 - 4.79i)T \) |
| good | 3 | \( 1 + (1.20 - 1.69i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (0.434 - 0.0836i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-1.43 + 1.13i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (5.14 - 3.30i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.17 - 2.07i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.28 - 3.12i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-5.02 + 1.47i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (4.34 - 0.415i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-1.87 - 5.42i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-0.691 - 0.798i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.17 + 2.58i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-4.73 + 8.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.413 - 8.69i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (4.04 + 2.08i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-3.67 - 5.15i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-5.77 + 2.31i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.992 + 6.90i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.18 - 9.01i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.738 + 15.4i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (6.05 - 6.98i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-11.5 - 1.10i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-2.48 - 2.87i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.76633086138363822965091729946, −10.92661008633282967097910074039, −10.12041093397429084399241211622, −9.379741240183005202026415858753, −8.252263294428800895330435541794, −7.31778795182492803945394223685, −5.81359139258596279880865715845, −4.71382197596615854238221953489, −3.83311461998334599799604067212, −1.90356205717438278296898512735,
0.62283545552422778513449272047, 2.22375586122036465257557292706, 4.58027470056095769581877635847, 5.57291038241009708838266604295, 6.73851236779914460113456514120, 7.47686832866592517790983028172, 8.170718632450148296480113212438, 9.432899133424135893182987334714, 10.49196135645599224550596187380, 11.41447229323990586249230131382
