L(s) = 1 | + (0.445 + 0.323i)3-s + (0.710 − 2.18i)5-s + (3.17 − 2.30i)7-s + (−0.833 − 2.56i)9-s + (−3.07 + 1.24i)11-s + (0.309 + 0.951i)13-s + (1.02 − 0.744i)15-s + (0.370 − 1.13i)17-s + (−5.34 − 3.88i)19-s + 2.16·21-s − 4.07·23-s + (−0.225 − 0.164i)25-s + (0.970 − 2.98i)27-s + (0.649 − 0.471i)29-s + (2.37 + 7.31i)31-s + ⋯ |
L(s) = 1 | + (0.257 + 0.187i)3-s + (0.317 − 0.977i)5-s + (1.19 − 0.871i)7-s + (−0.277 − 0.854i)9-s + (−0.927 + 0.374i)11-s + (0.0857 + 0.263i)13-s + (0.264 − 0.192i)15-s + (0.0898 − 0.276i)17-s + (−1.22 − 0.890i)19-s + 0.471·21-s − 0.850·23-s + (−0.0451 − 0.0328i)25-s + (0.186 − 0.574i)27-s + (0.120 − 0.0876i)29-s + (0.426 + 1.31i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36309 - 0.965726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36309 - 0.965726i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (3.07 - 1.24i)T \) |
| 13 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (-0.445 - 0.323i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.710 + 2.18i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-3.17 + 2.30i)T + (2.16 - 6.65i)T^{2} \) |
| 17 | \( 1 + (-0.370 + 1.13i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (5.34 + 3.88i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 4.07T + 23T^{2} \) |
| 29 | \( 1 + (-0.649 + 0.471i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.37 - 7.31i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.68 + 4.12i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.10 - 2.25i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + (-9.33 - 6.77i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.21 + 6.82i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.45 - 2.50i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.52 + 4.69i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 4.27T + 67T^{2} \) |
| 71 | \( 1 + (3.32 - 10.2i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.82 - 3.50i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.82 - 8.69i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.235 - 0.724i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 1.79T + 89T^{2} \) |
| 97 | \( 1 + (2.14 + 6.59i)T + (-78.4 + 57.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.60988708909814906898043622057, −9.579238965117021678213449373568, −8.772720225810602871730112217094, −8.066989970337542515063320963760, −7.10294170337029462917124108153, −5.83530337341737810175729878828, −4.72059941497367870997725040493, −4.18617426741825714454145328968, −2.45087030849985427802961478822, −0.959196513201054819678335032664,
2.10759658775407998160021818169, 2.65821414700322599507913530357, 4.33298865815151482707843706595, 5.58289382318098365319368744893, 6.13903993399391526355739577679, 7.76275737202106670690245169305, 7.992660718759005551336003722860, 8.968216518233081942355093211464, 10.46532251998340725142120312019, 10.66049116998554557795147958860