| L(s) = 1 | + (−0.415 + 0.909i)3-s + (−2.35 + 2.72i)5-s + (1.95 − 1.25i)7-s + (−0.654 − 0.755i)9-s + (−0.449 + 3.12i)11-s + (0.778 + 0.500i)13-s + (−1.49 − 3.27i)15-s + (−2.42 − 0.711i)17-s + (−4.25 + 1.24i)19-s + (0.330 + 2.29i)21-s + (−2.77 + 3.91i)23-s + (−1.13 − 7.89i)25-s + (0.959 − 0.281i)27-s + (−9.66 − 2.83i)29-s + (0.151 + 0.332i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 + 0.525i)3-s + (−1.05 + 1.21i)5-s + (0.737 − 0.473i)7-s + (−0.218 − 0.251i)9-s + (−0.135 + 0.942i)11-s + (0.215 + 0.138i)13-s + (−0.386 − 0.846i)15-s + (−0.587 − 0.172i)17-s + (−0.975 + 0.286i)19-s + (0.0720 + 0.500i)21-s + (−0.578 + 0.816i)23-s + (−0.227 − 1.57i)25-s + (0.184 − 0.0542i)27-s + (−1.79 − 0.527i)29-s + (0.0272 + 0.0596i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0724635 + 0.642220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0724635 + 0.642220i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (2.77 - 3.91i)T \) |
| good | 5 | \( 1 + (2.35 - 2.72i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.95 + 1.25i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.449 - 3.12i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.778 - 0.500i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (2.42 + 0.711i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (4.25 - 1.24i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (9.66 + 2.83i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.151 - 0.332i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (4.54 + 5.24i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.45 + 5.14i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (3.30 - 7.24i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 0.649T + 47T^{2} \) |
| 53 | \( 1 + (-4.44 + 2.85i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-11.5 - 7.44i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.28 - 9.38i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 5.62i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.930 - 6.47i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.41 + 1.88i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (0.637 + 0.409i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.06 - 5.84i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (6.23 - 13.6i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (1.28 - 1.48i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.16339840462097311879357785686, −10.53265090061546995919764859460, −9.651707745367797151550569479231, −8.404842586520486504362651524779, −7.47024081136642899566333080625, −6.93377153133059450170922339918, −5.62973006896740819659860883007, −4.24764336747434851501311012805, −3.82631249192034443091529027049, −2.22524230152950871022519494269,
0.37275734758072721038177405743, 1.93742933343701779515330351433, 3.72227351773235018832089274396, 4.78280979853809026819725900744, 5.60440778162288260560955293128, 6.77218613969349155488966249264, 8.095617465206509835615641189568, 8.343505541533360629323912715232, 9.116762607567487071685502497352, 10.74135624522550383187448205845
