L(s) = 1 | − i·2-s + (−1.36 − 1.06i)3-s − 4-s − i·5-s + (−1.06 + 1.36i)6-s + 1.28·7-s + i·8-s + (0.744 + 2.90i)9-s − 10-s − 0.392·11-s + (1.36 + 1.06i)12-s − 4.84i·13-s − 1.28i·14-s + (−1.06 + 1.36i)15-s + 16-s − 6.52·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.789 − 0.613i)3-s − 0.5·4-s − 0.447i·5-s + (−0.433 + 0.558i)6-s + 0.484·7-s + 0.353i·8-s + (0.248 + 0.968i)9-s − 0.316·10-s − 0.118·11-s + (0.394 + 0.306i)12-s − 1.34i·13-s − 0.342i·14-s + (−0.274 + 0.353i)15-s + 0.250·16-s − 1.58·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.145762 + 0.200577i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.145762 + 0.200577i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.36 + 1.06i)T \) |
| 5 | \( 1 + iT \) |
| 31 | \( 1 + (-4.60 - 3.12i)T \) |
good | 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 + 0.392T + 11T^{2} \) |
| 13 | \( 1 + 4.84iT - 13T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 + 4.57T + 29T^{2} \) |
| 37 | \( 1 - 2.12iT - 37T^{2} \) |
| 41 | \( 1 - 5.07iT - 41T^{2} \) |
| 43 | \( 1 - 3.28iT - 43T^{2} \) |
| 47 | \( 1 + 0.834iT - 47T^{2} \) |
| 53 | \( 1 + 0.123T + 53T^{2} \) |
| 59 | \( 1 + 5.86iT - 59T^{2} \) |
| 61 | \( 1 - 9.87iT - 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 9.33iT - 71T^{2} \) |
| 73 | \( 1 + 9.18iT - 73T^{2} \) |
| 79 | \( 1 + 6.12iT - 79T^{2} \) |
| 83 | \( 1 + 2.73T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−9.700482047503171540628931284117, −8.432453970463478445079612136166, −8.080617894587718289441336047777, −6.83902465540805828004600310884, −5.88972670015559184343890541234, −4.98311543421190730297574415138, −4.24416338523248538439210069283, −2.62395659035950811900704621461, −1.53968369823339887081159236333, −0.12730051645729897001040767669,
2.09935842095465783821110011997, 3.99756204583018415746730662261, 4.44662815258800435054358962601, 5.51973294352888063838123617924, 6.52433676730418865005396236216, 6.87356357498279636443595992559, 8.158049987308265185112358672375, 9.038343713470531216139504562969, 9.711729184689478602807384226904, 10.79313141345558298320448186120