| L(s) = 1 | + (−1.41 − 0.0706i)2-s + (−1.01 + 2.89i)3-s + (1.99 + 0.199i)4-s + (−1.69 − 0.387i)5-s + (1.63 − 4.01i)6-s + (0.668 + 1.38i)7-s + (−2.79 − 0.422i)8-s + (−5.00 − 3.99i)9-s + (2.37 + 0.667i)10-s + (−4.29 − 0.483i)11-s + (−2.59 + 5.55i)12-s + (−0.896 + 0.714i)13-s + (−0.846 − 2.00i)14-s + (2.84 − 4.52i)15-s + (3.92 + 0.794i)16-s + (5.08 + 5.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.0499i)2-s + (−0.584 + 1.67i)3-s + (0.995 + 0.0998i)4-s + (−0.759 − 0.173i)5-s + (0.667 − 1.63i)6-s + (0.252 + 0.524i)7-s + (−0.988 − 0.149i)8-s + (−1.66 − 1.33i)9-s + (0.749 + 0.211i)10-s + (−1.29 − 0.145i)11-s + (−0.748 + 1.60i)12-s + (−0.248 + 0.198i)13-s + (−0.226 − 0.536i)14-s + (0.733 − 1.16i)15-s + (0.980 + 0.198i)16-s + (1.23 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0483675 + 0.354894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0483675 + 0.354894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0706i)T \) |
| 29 | \( 1 + (-5.15 - 1.56i)T \) |
| good | 3 | \( 1 + (1.01 - 2.89i)T + (-2.34 - 1.87i)T^{2} \) |
| 5 | \( 1 + (1.69 + 0.387i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (-0.668 - 1.38i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (4.29 + 0.483i)T + (10.7 + 2.44i)T^{2} \) |
| 13 | \( 1 + (0.896 - 0.714i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-5.08 - 5.08i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.01 - 1.05i)T + (14.8 - 11.8i)T^{2} \) |
| 23 | \( 1 + (-1.23 + 0.281i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (-4.54 - 7.23i)T + (-13.4 + 27.9i)T^{2} \) |
| 37 | \( 1 + (-0.587 - 5.21i)T + (-36.0 + 8.23i)T^{2} \) |
| 41 | \( 1 + (3.03 - 3.03i)T - 41iT^{2} \) |
| 43 | \( 1 + (1.81 + 1.13i)T + (18.6 + 38.7i)T^{2} \) |
| 47 | \( 1 + (0.424 - 3.76i)T + (-45.8 - 10.4i)T^{2} \) |
| 53 | \( 1 + (1.66 - 7.29i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + 1.69iT - 59T^{2} \) |
| 61 | \( 1 + (-3.78 - 1.32i)T + (47.6 + 38.0i)T^{2} \) |
| 67 | \( 1 + (-2.99 + 3.76i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (1.15 + 1.45i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (4.80 + 3.01i)T + (31.6 + 65.7i)T^{2} \) |
| 79 | \( 1 + (1.54 + 13.6i)T + (-77.0 + 17.5i)T^{2} \) |
| 83 | \( 1 + (-3.22 + 6.68i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-4.41 + 2.77i)T + (38.6 - 80.1i)T^{2} \) |
| 97 | \( 1 + (4.80 - 1.68i)T + (75.8 - 60.4i)T^{2} \) |
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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
−14.68425835094816783389763989717, −12.37501057766245951195528316022, −11.65390839437473366060532990623, −10.46502348161538775427927122889, −10.18943290294795483286082161721, −8.727987335867085018247812275311, −8.016633217522553599396032838599, −6.05995985913261091669443398348, −4.83764676761189829156696609571, −3.24602292950482018031494722803,
0.55603937932678862241448766879, 2.55108468219811963042248379960, 5.48195850641892218829774158606, 6.90146402292536647155576072014, 7.64792531606135044627228540817, 8.150747822190842621856594087090, 10.07517058392741956751356810043, 11.19614831484516476104506498478, 11.87521535449153739799984721025, 12.79874369988464279521744392107
