| L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (0.411 − 2.86i)5-s + (−0.841 + 0.540i)6-s + (0.415 + 0.909i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (−1.20 + 2.62i)10-s + (0.803 − 0.235i)11-s + (0.959 − 0.281i)12-s + (−0.152 + 0.334i)13-s + (−0.142 − 0.989i)14-s + (−1.89 − 2.18i)15-s + (0.415 + 0.909i)16-s + (4.13 − 2.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.378 − 0.436i)3-s + (0.420 + 0.270i)4-s + (0.184 − 1.27i)5-s + (−0.343 + 0.220i)6-s + (0.157 + 0.343i)7-s + (−0.231 − 0.267i)8-s + (−0.0474 − 0.329i)9-s + (−0.379 + 0.831i)10-s + (0.242 − 0.0711i)11-s + (0.276 − 0.0813i)12-s + (−0.0423 + 0.0926i)13-s + (−0.0380 − 0.264i)14-s + (−0.488 − 0.564i)15-s + (0.103 + 0.227i)16-s + (1.00 − 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.618431 - 1.11224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.618431 - 1.11224i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.61 + 1.31i)T \) |
| good | 5 | \( 1 + (-0.411 + 2.86i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-0.803 + 0.235i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.152 - 0.334i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.13 + 2.66i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (5.60 + 3.60i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.968 + 0.622i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.454 - 0.524i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (1.06 + 7.39i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.696 + 4.84i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (3.87 - 4.47i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + (-3.90 - 8.54i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.451 - 0.989i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-2.54 - 2.93i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-3.47 - 1.02i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (0.514 + 0.151i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.30 + 2.12i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.69 + 8.09i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.62 + 11.2i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.70 + 5.43i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.00 - 6.99i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−9.374280299453960780797983578232, −8.932608902835840679616174884104, −8.330430546278284523616759175299, −7.42457601141842879330020399269, −6.48380129390458490053100751714, −5.37289180834114095671645834681, −4.44406194853488551785139249526, −3.01902192023243383973614742849, −1.85447371442099374573442293995, −0.72495475443258319772013014377,
1.68071744888507953010920539500, 2.95500446548718854619975023176, 3.79711032589368381983102112332, 5.17069689898152322953289253632, 6.37180322635401595684210164373, 6.88880592484558545757442568317, 7.981505912064465205563887945099, 8.472577780106945551092086952109, 9.764067298471760632819425808377, 10.12833944936638426012987452713
