| L(s) = 1 | + (−0.815 − 0.274i)2-s + (−1.00 − 0.761i)4-s + (−1.45 + 3.65i)5-s + (0.847 − 0.392i)7-s + (1.57 + 2.32i)8-s + (2.19 − 2.58i)10-s + (1.30 − 4.69i)11-s + (−1.78 + 3.36i)13-s + (−0.799 + 0.0869i)14-s + (0.0273 + 0.0986i)16-s + (−4.49 − 2.07i)17-s + (−1.47 + 0.325i)19-s + (4.24 − 2.55i)20-s + (−2.35 + 3.47i)22-s + (0.497 − 3.03i)23-s + ⋯ |
| L(s) = 1 | + (−0.576 − 0.194i)2-s + (−0.501 − 0.380i)4-s + (−0.651 + 1.63i)5-s + (0.320 − 0.148i)7-s + (0.556 + 0.820i)8-s + (0.693 − 0.816i)10-s + (0.393 − 1.41i)11-s + (−0.494 + 0.932i)13-s + (−0.213 + 0.0232i)14-s + (0.00684 + 0.0246i)16-s + (−1.08 − 0.504i)17-s + (−0.339 + 0.0746i)19-s + (0.948 − 0.570i)20-s + (−0.502 + 0.740i)22-s + (0.103 − 0.632i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000774359 - 0.0108386i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000774359 - 0.0108386i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-4.84 + 5.95i)T \) |
| good | 2 | \( 1 + (0.815 + 0.274i)T + (1.59 + 1.21i)T^{2} \) |
| 5 | \( 1 + (1.45 - 3.65i)T + (-3.62 - 3.43i)T^{2} \) |
| 7 | \( 1 + (-0.847 + 0.392i)T + (4.53 - 5.33i)T^{2} \) |
| 11 | \( 1 + (-1.30 + 4.69i)T + (-9.42 - 5.67i)T^{2} \) |
| 13 | \( 1 + (1.78 - 3.36i)T + (-7.29 - 10.7i)T^{2} \) |
| 17 | \( 1 + (4.49 + 2.07i)T + (11.0 + 12.9i)T^{2} \) |
| 19 | \( 1 + (1.47 - 0.325i)T + (17.2 - 7.97i)T^{2} \) |
| 23 | \( 1 + (-0.497 + 3.03i)T + (-21.7 - 7.34i)T^{2} \) |
| 29 | \( 1 + (8.77 - 2.95i)T + (23.0 - 17.5i)T^{2} \) |
| 31 | \( 1 + (7.13 + 1.57i)T + (28.1 + 13.0i)T^{2} \) |
| 37 | \( 1 + (-2.45 + 3.62i)T + (-13.6 - 34.3i)T^{2} \) |
| 41 | \( 1 + (-1.69 - 10.3i)T + (-38.8 + 13.0i)T^{2} \) |
| 43 | \( 1 + (0.416 + 1.49i)T + (-36.8 + 22.1i)T^{2} \) |
| 47 | \( 1 + (2.56 + 6.42i)T + (-34.1 + 32.3i)T^{2} \) |
| 53 | \( 1 + (-0.552 - 0.650i)T + (-8.57 + 52.3i)T^{2} \) |
| 61 | \( 1 + (2.47 + 0.834i)T + (48.5 + 36.9i)T^{2} \) |
| 67 | \( 1 + (4.60 + 6.79i)T + (-24.7 + 62.2i)T^{2} \) |
| 71 | \( 1 + (0.745 + 1.87i)T + (-51.5 + 48.8i)T^{2} \) |
| 73 | \( 1 + (10.4 - 1.13i)T + (71.2 - 15.6i)T^{2} \) |
| 79 | \( 1 + (-3.21 + 1.93i)T + (37.0 - 69.7i)T^{2} \) |
| 83 | \( 1 + (-0.128 - 2.36i)T + (-82.5 + 8.97i)T^{2} \) |
| 89 | \( 1 + (4.43 - 1.49i)T + (70.8 - 53.8i)T^{2} \) |
| 97 | \( 1 + (7.49 + 0.814i)T + (94.7 + 20.8i)T^{2} \) |
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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
−11.22755424235733706668522500667, −10.64372028475758773445450689439, −9.503818574765068875894215382269, −8.776945128750122852000666517217, −7.75912228486554752645982047356, −6.89875675245833036442305306455, −5.97222853955194151528562802900, −4.53051896946693824808017968873, −3.49903354013838128681748876116, −2.12232769203048611433297918719,
0.00774377144225525343398097197, 1.68521841529544973600847495478, 3.92657200760906534615967486993, 4.55780358635098846369392847328, 5.45382297467040293877558114055, 7.24513250059421933195292905298, 7.78180312137108823807821224597, 8.725017378035971536902235869499, 9.200027063115971622026513540253, 10.04524246890190568126420858624
