L(s) = 1 | + (0.382 + 0.923i)3-s + (0.923 − 0.382i)4-s + (−0.785 + 0.785i)7-s + (−0.707 + 0.707i)9-s + (0.707 + 0.707i)12-s + (−0.168 + 0.555i)13-s + (0.707 − 0.707i)16-s + (−0.151 − 1.53i)19-s + (−1.02 − 0.425i)21-s + (0.980 − 0.195i)25-s + (−0.923 − 0.382i)27-s + (−0.425 + 1.02i)28-s + (−0.360 − 1.81i)31-s + (−0.382 + 0.923i)36-s + (−0.831 + 0.444i)37-s + ⋯ |
L(s) = 1 | + (0.382 + 0.923i)3-s + (0.923 − 0.382i)4-s + (−0.785 + 0.785i)7-s + (−0.707 + 0.707i)9-s + (0.707 + 0.707i)12-s + (−0.168 + 0.555i)13-s + (0.707 − 0.707i)16-s + (−0.151 − 1.53i)19-s + (−1.02 − 0.425i)21-s + (0.980 − 0.195i)25-s + (−0.923 − 0.382i)27-s + (−0.425 + 1.02i)28-s + (−0.360 − 1.81i)31-s + (−0.382 + 0.923i)36-s + (−0.831 + 0.444i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.117023626\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.117023626\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.382 - 0.923i)T \) |
| 193 | \( 1 + (-0.923 + 0.382i)T \) |
good | 2 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 5 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 7 | \( 1 + (0.785 - 0.785i)T - iT^{2} \) |
| 11 | \( 1 + (0.195 - 0.980i)T^{2} \) |
| 13 | \( 1 + (0.168 - 0.555i)T + (-0.831 - 0.555i)T^{2} \) |
| 17 | \( 1 + (-0.980 - 0.195i)T^{2} \) |
| 19 | \( 1 + (0.151 + 1.53i)T + (-0.980 + 0.195i)T^{2} \) |
| 23 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (-0.555 - 0.831i)T^{2} \) |
| 31 | \( 1 + (0.360 + 1.81i)T + (-0.923 + 0.382i)T^{2} \) |
| 37 | \( 1 + (0.831 - 0.444i)T + (0.555 - 0.831i)T^{2} \) |
| 41 | \( 1 + (0.195 - 0.980i)T^{2} \) |
| 43 | \( 1 + (-0.275 - 0.275i)T + iT^{2} \) |
| 47 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 53 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.195 - 1.98i)T + (-0.980 - 0.195i)T^{2} \) |
| 67 | \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 71 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 73 | \( 1 + (0.172 + 0.0924i)T + (0.555 + 0.831i)T^{2} \) |
| 79 | \( 1 + (1.36 - 1.11i)T + (0.195 - 0.980i)T^{2} \) |
| 83 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 89 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 97 | \( 1 + (1.38 + 0.275i)T + (0.923 + 0.382i)T^{2} \) |
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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
−11.03616533230634314254705053846, −10.11394458640604484480666611988, −9.356053700171297330404133393199, −8.721620724252098461565045333940, −7.40834533323777937334464258602, −6.44922142637906038565225069640, −5.57167280303546846736267924476, −4.49841525502332942481930808319, −3.06965488896044187763369207869, −2.34148875325311238792463247072,
1.56142756919976868188868371924, 2.98152666404592363843685980755, 3.67910555656296836693685393571, 5.60015983535386887225549326769, 6.62236141925963446401248072249, 7.14851882433413638912147642428, 7.963549538783921277104210858340, 8.825657968990524181847410781684, 10.14896324551784165091859981989, 10.74127009919836613440545597164