L(s) = 1 | + (0.861 + 1.50i)3-s + (1.19 − 1.19i)5-s − 7-s + (−1.51 + 2.58i)9-s + (−1.10 − 1.10i)11-s + (0.418 − 0.418i)13-s + (2.83 + 0.769i)15-s + 4.01i·17-s + (4.91 + 4.91i)19-s + (−0.861 − 1.50i)21-s − 3.26i·23-s + 2.12i·25-s + (−5.19 − 0.0508i)27-s + (4.61 + 4.61i)29-s + 7.31i·31-s + ⋯ |
L(s) = 1 | + (0.497 + 0.867i)3-s + (0.536 − 0.536i)5-s − 0.377·7-s + (−0.505 + 0.862i)9-s + (−0.332 − 0.332i)11-s + (0.116 − 0.116i)13-s + (0.731 + 0.198i)15-s + 0.974i·17-s + (1.12 + 1.12i)19-s + (−0.187 − 0.327i)21-s − 0.681i·23-s + 0.425i·25-s + (−0.999 − 0.00979i)27-s + (0.857 + 0.857i)29-s + 1.31i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.969666580\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.969666580\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.861 - 1.50i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-1.19 + 1.19i)T - 5iT^{2} \) |
| 11 | \( 1 + (1.10 + 1.10i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.418 + 0.418i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.01iT - 17T^{2} \) |
| 19 | \( 1 + (-4.91 - 4.91i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.26iT - 23T^{2} \) |
| 29 | \( 1 + (-4.61 - 4.61i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.31iT - 31T^{2} \) |
| 37 | \( 1 + (-2.61 - 2.61i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.46T + 41T^{2} \) |
| 43 | \( 1 + (3.91 - 3.91i)T - 43iT^{2} \) |
| 47 | \( 1 - 8.08T + 47T^{2} \) |
| 53 | \( 1 + (-1.78 + 1.78i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.55 + 5.55i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.325 + 0.325i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.41 - 1.41i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 + 2.89iT - 73T^{2} \) |
| 79 | \( 1 + 15.6iT - 79T^{2} \) |
| 83 | \( 1 + (9.34 - 9.34i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 3.43T + 97T^{2} \) |
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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
−9.787193481480361441772288041304, −8.998584793276416941462205810719, −8.392987779059130034686514934214, −7.58639896847769217300021827601, −6.26921270271551262086609280527, −5.50862195534927114864024819093, −4.74140014186049701017763173570, −3.64031193733907615819340192210, −2.87111930109845668511974560542, −1.45011178888582168149035983356,
0.809284716332509335805928050150, 2.42643034290320295858838760052, 2.81737268682986666686559190951, 4.16630280632403876030653409030, 5.50132450708711904322412803201, 6.23793028192970768888198519904, 7.18680631183134834791386093063, 7.52209674943379347072721031638, 8.660049774430664406981258223503, 9.524214835172354154084784864162