| 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
−12.79874369988464279521744392107, −11.87521535449153739799984721025, −11.19614831484516476104506498478, −10.07517058392741956751356810043, −8.150747822190842621856594087090, −7.64792531606135044627228540817, −6.90146402292536647155576072014, −5.48195850641892218829774158606, −2.55108468219811963042248379960, −0.55603937932678862241448766879,
3.24602292950482018031494722803, 4.83764676761189829156696609571, 6.05995985913261091669443398348, 8.016633217522553599396032838599, 8.727987335867085018247812275311, 10.18943290294795483286082161721, 10.46502348161538775427927122889, 11.65390839437473366060532990623, 12.37501057766245951195528316022, 14.68425835094816783389763989717
