L(s) = 1 | + (0.858 − 0.512i)2-s + (0.0518 + 0.263i)3-s + (0.473 − 0.880i)4-s + (0.179 + 0.199i)6-s + (−0.0448 − 0.998i)8-s + (0.858 − 0.351i)9-s + (0.193 + 0.981i)11-s + (0.256 + 0.0791i)12-s + (−0.550 − 0.834i)16-s + (−0.470 + 0.940i)17-s + (0.556 − 0.742i)18-s + (1.29 − 1.52i)19-s + (0.669 + 0.743i)22-s + (0.260 − 0.0636i)24-s + (−0.946 − 0.323i)25-s + ⋯ |
L(s) = 1 | + (0.858 − 0.512i)2-s + (0.0518 + 0.263i)3-s + (0.473 − 0.880i)4-s + (0.179 + 0.199i)6-s + (−0.0448 − 0.998i)8-s + (0.858 − 0.351i)9-s + (0.193 + 0.981i)11-s + (0.256 + 0.0791i)12-s + (−0.550 − 0.834i)16-s + (−0.470 + 0.940i)17-s + (0.556 − 0.742i)18-s + (1.29 − 1.52i)19-s + (0.669 + 0.743i)22-s + (0.260 − 0.0636i)24-s + (−0.946 − 0.323i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.382118096\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.382118096\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.858 + 0.512i)T \) |
| 11 | \( 1 + (-0.193 - 0.981i)T \) |
| 43 | \( 1 + (-0.420 - 0.907i)T \) |
good | 3 | \( 1 + (-0.0518 - 0.263i)T + (-0.925 + 0.379i)T^{2} \) |
| 5 | \( 1 + (0.946 + 0.323i)T^{2} \) |
| 7 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 13 | \( 1 + (-0.992 - 0.119i)T^{2} \) |
| 17 | \( 1 + (0.470 - 0.940i)T + (-0.599 - 0.800i)T^{2} \) |
| 19 | \( 1 + (-1.29 + 1.52i)T + (-0.163 - 0.986i)T^{2} \) |
| 23 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (0.925 + 0.379i)T^{2} \) |
| 31 | \( 1 + (0.999 - 0.0299i)T^{2} \) |
| 37 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 41 | \( 1 + (-0.607 + 1.42i)T + (-0.691 - 0.722i)T^{2} \) |
| 47 | \( 1 + (-0.936 - 0.351i)T^{2} \) |
| 53 | \( 1 + (-0.193 + 0.981i)T^{2} \) |
| 59 | \( 1 + (0.0890 - 1.98i)T + (-0.995 - 0.0896i)T^{2} \) |
| 61 | \( 1 + (0.999 + 0.0299i)T^{2} \) |
| 67 | \( 1 + (0.0156 - 0.208i)T + (-0.988 - 0.149i)T^{2} \) |
| 71 | \( 1 + (-0.251 + 0.967i)T^{2} \) |
| 73 | \( 1 + (0.492 + 1.89i)T + (-0.873 + 0.486i)T^{2} \) |
| 79 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 83 | \( 1 + (1.49 - 0.836i)T + (0.525 - 0.850i)T^{2} \) |
| 89 | \( 1 + (1.46 + 1.35i)T + (0.0747 + 0.997i)T^{2} \) |
| 97 | \( 1 + (-0.112 + 0.833i)T + (-0.963 - 0.266i)T^{2} \) |
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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
−8.888202691562550171622210304706, −7.43277039409463119928186389014, −7.14934152036317885037365295975, −6.21763511113718750083730355883, −5.46454118726153850379802520137, −4.36984340668981035804892443523, −4.28529442646546700119200020542, −3.14946894133317406338871861595, −2.20906860680838934457705628632, −1.21661902787763290125618463602,
1.44130187651474095221300513352, 2.57458558166132987180330246160, 3.53336331062227905124155724198, 4.15955118015058229334500899483, 5.18016216805766680847525533862, 5.72812136329656610775768843258, 6.54782027508218952243676083869, 7.29002403943102221660452050337, 7.83778571038937225633963040027, 8.465776877649414580006750046497