L(s) = 1 | + (−2.41 + 1.47i)2-s − 4.79·3-s + (3.65 − 7.11i)4-s − 17.0i·5-s + (11.5 − 7.07i)6-s + (−18.3 + 2.33i)7-s + (1.65 + 22.5i)8-s − 3.97·9-s + (25.1 + 41.2i)10-s − 41.4i·11-s + (−17.5 + 34.1i)12-s + 45.3i·13-s + (40.9 − 32.7i)14-s + 81.9i·15-s + (−37.2 − 52.0i)16-s − 28.2i·17-s + ⋯ |
L(s) = 1 | + (−0.853 + 0.521i)2-s − 0.923·3-s + (0.457 − 0.889i)4-s − 1.52i·5-s + (0.788 − 0.481i)6-s + (−0.992 + 0.126i)7-s + (0.0732 + 0.997i)8-s − 0.147·9-s + (0.795 + 1.30i)10-s − 1.13i·11-s + (−0.422 + 0.821i)12-s + 0.967i·13-s + (0.780 − 0.624i)14-s + 1.41i·15-s + (−0.582 − 0.813i)16-s − 0.403i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.202196 - 0.288508i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.202196 - 0.288508i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.41 - 1.47i)T \) |
| 7 | \( 1 + (18.3 - 2.33i)T \) |
good | 3 | \( 1 + 4.79T + 27T^{2} \) |
| 5 | \( 1 + 17.0iT - 125T^{2} \) |
| 11 | \( 1 + 41.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 45.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 28.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 41.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 93.9iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 27.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 81.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 171. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 286.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 575.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 778. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 198. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 197. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 255. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 938.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.16e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 656. iT - 9.12e5T^{2} \) |
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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
−16.41183480075445137120861314468, −16.05264601618231956958750666542, −13.91193915666030946553154950966, −12.35029038093046743946053563722, −11.18521490040208814813119857758, −9.483545097032289637365992027721, −8.527013191567073963035902751413, −6.43689576152364603152117054697, −5.25128198889239571599860687932, −0.48382753337899714939404301192,
3.09369947912311861361672723651, 6.31057667914380707099657186614, 7.46585134175542902233422835113, 9.790138308757607749079305026429, 10.60168312811459693735507330291, 11.68449623198371856810630185470, 12.96023748626862339527195958103, 14.95206882876036865769447119380, 16.17642971969142210940696808780, 17.56319348656835794269504317774