| 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
−10.74135624522550383187448205845, −9.116762607567487071685502497352, −8.343505541533360629323912715232, −8.095617465206509835615641189568, −6.77218613969349155488966249264, −5.60440778162288260560955293128, −4.78280979853809026819725900744, −3.72227351773235018832089274396, −1.93742933343701779515330351433, −0.37275734758072721038177405743,
2.22524230152950871022519494269, 3.82631249192034443091529027049, 4.24764336747434851501311012805, 5.62973006896740819659860883007, 6.93377153133059450170922339918, 7.47024081136642899566333080625, 8.404842586520486504362651524779, 9.651707745367797151550569479231, 10.53265090061546995919764859460, 11.16339840462097311879357785686
