| L(s) = 1 | + (−0.993 − 0.478i)2-s + (−0.488 − 0.612i)4-s + (3.17 + 0.979i)5-s + (1.23 + 2.13i)7-s + (0.683 + 2.99i)8-s + (−2.68 − 2.49i)10-s + (−0.748 + 0.937i)11-s + (−4.22 + 3.91i)13-s + (−0.203 − 2.71i)14-s + (0.404 − 1.77i)16-s + (−1.02 + 0.314i)17-s + (2.13 − 0.321i)19-s + (−0.951 − 2.42i)20-s + (1.19 − 0.574i)22-s + (2.16 + 5.50i)23-s + ⋯ |
| L(s) = 1 | + (−0.702 − 0.338i)2-s + (−0.244 − 0.306i)4-s + (1.41 + 0.437i)5-s + (0.465 + 0.807i)7-s + (0.241 + 1.05i)8-s + (−0.849 − 0.788i)10-s + (−0.225 + 0.282i)11-s + (−1.17 + 1.08i)13-s + (−0.0542 − 0.724i)14-s + (0.101 − 0.442i)16-s + (−0.247 + 0.0763i)17-s + (0.489 − 0.0737i)19-s + (−0.212 − 0.542i)20-s + (0.254 − 0.122i)22-s + (0.450 + 1.14i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02594 + 0.265422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02594 + 0.265422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (2.45 + 6.08i)T \) |
| good | 2 | \( 1 + (0.993 + 0.478i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (-3.17 - 0.979i)T + (4.13 + 2.81i)T^{2} \) |
| 7 | \( 1 + (-1.23 - 2.13i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.748 - 0.937i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (4.22 - 3.91i)T + (0.971 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.02 - 0.314i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.13 + 0.321i)T + (18.1 - 5.60i)T^{2} \) |
| 23 | \( 1 + (-2.16 - 5.50i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (0.177 + 2.37i)T + (-28.6 + 4.32i)T^{2} \) |
| 31 | \( 1 + (-6.31 + 4.30i)T + (11.3 - 28.8i)T^{2} \) |
| 37 | \( 1 + (2.83 - 4.90i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.77 - 4.22i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (3.30 + 4.14i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (2.69 + 2.50i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (0.208 - 0.914i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (2.31 + 1.57i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (-11.8 + 1.78i)T + (64.0 - 19.7i)T^{2} \) |
| 71 | \( 1 + (-5.54 + 14.1i)T + (-52.0 - 48.2i)T^{2} \) |
| 73 | \( 1 + (4.99 - 4.63i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (1.18 + 2.05i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.533 + 7.11i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (-0.113 + 1.51i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (6.73 - 8.44i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−11.29297109563291839337604550499, −10.15408451685606140638866835206, −9.607980799275257927538045461333, −9.078563262452401811723960934391, −7.87601157774507959295956466699, −6.60804254386900245069437556422, −5.52930338319139233245086309227, −4.81104423378735217183414485634, −2.50092301588564127909044915326, −1.78470756201342889083503536915,
0.957278861119717071059756055013, 2.77911934599765919510046615248, 4.51364956285583305519081785731, 5.42046880445073162858956936548, 6.71842859527599256761327063556, 7.65768993556091829794769332515, 8.502466430775812664435680023849, 9.455425928275535722140863340088, 10.11133131131109717433712005140, 10.82533963416598778379036894615
