L(s) = 1 | + (0.860 − 0.625i)2-s + (1.72 + 0.197i)3-s + (−0.268 + 0.825i)4-s + (0.587 − 0.809i)5-s + (1.60 − 0.905i)6-s + (−2.18 − 0.709i)7-s + (0.942 + 2.90i)8-s + (2.92 + 0.681i)9-s − 1.06i·10-s + (−2.26 − 2.41i)11-s + (−0.625 + 1.36i)12-s + (−1.13 − 1.55i)13-s + (−2.32 + 0.754i)14-s + (1.17 − 1.27i)15-s + (1.22 + 0.887i)16-s + (3.01 + 2.19i)17-s + ⋯ |
L(s) = 1 | + (0.608 − 0.442i)2-s + (0.993 + 0.114i)3-s + (−0.134 + 0.412i)4-s + (0.262 − 0.361i)5-s + (0.655 − 0.369i)6-s + (−0.825 − 0.268i)7-s + (0.333 + 1.02i)8-s + (0.973 + 0.227i)9-s − 0.336i·10-s + (−0.683 − 0.729i)11-s + (−0.180 + 0.394i)12-s + (−0.313 − 0.431i)13-s + (−0.620 + 0.201i)14-s + (0.302 − 0.329i)15-s + (0.305 + 0.221i)16-s + (0.732 + 0.532i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.83012 - 0.215548i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83012 - 0.215548i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 - 0.197i)T \) |
| 5 | \( 1 + (-0.587 + 0.809i)T \) |
| 11 | \( 1 + (2.26 + 2.41i)T \) |
good | 2 | \( 1 + (-0.860 + 0.625i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (2.18 + 0.709i)T + (5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (1.13 + 1.55i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.01 - 2.19i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (8.13 - 2.64i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 4.81iT - 23T^{2} \) |
| 29 | \( 1 + (-1.01 + 3.13i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.98 + 2.89i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (3.59 - 11.0i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.619 - 1.90i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 6.20iT - 43T^{2} \) |
| 47 | \( 1 + (-3.15 + 1.02i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.55 - 4.88i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.0820 + 0.0266i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.79 + 6.60i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 3.92T + 67T^{2} \) |
| 71 | \( 1 + (-8.64 + 11.9i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-5.23 - 1.70i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.0325 - 0.0448i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.22 + 0.886i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 1.65iT - 89T^{2} \) |
| 97 | \( 1 + (-6.60 + 4.80i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.93872743160314255270487575216, −12.29645156698545987608212330199, −10.65251840357805879505672740715, −9.896257453203769170813407177181, −8.439641732151286166040821812609, −8.046704991581543433454653743963, −6.30408058921326295508296776307, −4.68520078825825901320454907858, −3.55519451897410230983832275642, −2.48863486572687035178389793690,
2.34813469759640710173394070298, 3.87114309592862183370809104655, 5.22985383632579298520119451053, 6.62431424693125182087986098445, 7.32388826736911380720761509997, 8.917696309163984007538675523513, 9.781940439246255544499635001628, 10.52206641330420614795322990086, 12.45009332048400627123217435550, 13.05019144077489566817369008621