L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.866 − 1.5i)3-s + (0.999 + 1.73i)4-s + (−4.67 − 1.76i)5-s − 2.44i·6-s + (−6.18 − 3.28i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−4.48 − 5.46i)10-s + (−1.46 − 2.53i)11-s + (1.73 − 2.99i)12-s − 16.5·13-s + (−5.25 − 8.39i)14-s + (1.40 + 8.54i)15-s + (−2.00 + 3.46i)16-s + (−10.2 − 17.6i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s + (−0.935 − 0.352i)5-s − 0.408i·6-s + (−0.883 − 0.468i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.448 − 0.546i)10-s + (−0.133 − 0.230i)11-s + (0.144 − 0.249i)12-s − 1.27·13-s + (−0.375 − 0.599i)14-s + (0.0939 + 0.569i)15-s + (−0.125 + 0.216i)16-s + (−0.600 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.148822 - 0.484811i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.148822 - 0.484811i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 + (4.67 + 1.76i)T \) |
| 7 | \( 1 + (6.18 + 3.28i)T \) |
good | 11 | \( 1 + (1.46 + 2.53i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 16.5T + 169T^{2} \) |
| 17 | \( 1 + (10.2 + 17.6i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-5.28 - 3.05i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (12.3 + 7.11i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 34.1T + 841T^{2} \) |
| 31 | \( 1 + (4.16 - 2.40i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (38.4 + 22.1i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 39.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-2.08 + 3.60i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-51.1 + 29.5i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-31.2 + 18.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (47.8 + 27.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-60.0 + 34.6i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 110.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-47.6 - 82.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (64.7 - 112. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 5.92T + 6.88e3T^{2} \) |
| 89 | \( 1 + (85.9 + 49.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 116.T + 9.40e3T^{2} \) |
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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
−12.12200755487028064481598745078, −11.08692793284011851467533770337, −9.815096834629850176072269929846, −8.475092102679473060330925849028, −7.33701820268237758884014602602, −6.80097463575852004699868135833, −5.33586940927297695582528708863, −4.24668429221992835479115725133, −2.84741128300323723911979552840, −0.22651969815885994225396648235,
2.67712910085895788236615064373, 3.84687572448504268970011981421, 4.92934981655044846996789569308, 6.23957367171338556885998090340, 7.25571562223533778999517744981, 8.695658170929933679564796663275, 9.959910632227894690749004794527, 10.58788512336869408324992595430, 11.89656969270992902321304979417, 12.19291270532159533239545164931