| L(s) = 1 | + (24.5 + 29.2i)3-s + (−71.7 − 26.1i)5-s + (38.3 − 66.4i)7-s + (−127. + 721. i)9-s + (1.03e3 + 1.79e3i)11-s + (−2.34e3 + 2.79e3i)13-s + (−998. − 2.74e3i)15-s + (492. + 2.79e3i)17-s + (−5.78e3 − 3.67e3i)19-s + (2.88e3 − 509. i)21-s + (−7.82e3 + 2.84e3i)23-s + (−7.50e3 − 6.29e3i)25-s + (−131. + 75.8i)27-s + (−7.79e3 − 1.37e3i)29-s + (−7.94e3 − 4.58e3i)31-s + ⋯ |
| L(s) = 1 | + (0.910 + 1.08i)3-s + (−0.574 − 0.208i)5-s + (0.111 − 0.193i)7-s + (−0.174 + 0.990i)9-s + (0.776 + 1.34i)11-s + (−1.06 + 1.27i)13-s + (−0.295 − 0.812i)15-s + (0.100 + 0.568i)17-s + (−0.844 − 0.536i)19-s + (0.311 − 0.0549i)21-s + (−0.643 + 0.234i)23-s + (−0.480 − 0.402i)25-s + (−0.00667 + 0.00385i)27-s + (−0.319 − 0.0563i)29-s + (−0.266 − 0.154i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.610493 + 1.62881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.610493 + 1.62881i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (5.78e3 + 3.67e3i)T \) |
| good | 3 | \( 1 + (-24.5 - 29.2i)T + (-126. + 717. i)T^{2} \) |
| 5 | \( 1 + (71.7 + 26.1i)T + (1.19e4 + 1.00e4i)T^{2} \) |
| 7 | \( 1 + (-38.3 + 66.4i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.03e3 - 1.79e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (2.34e3 - 2.79e3i)T + (-8.38e5 - 4.75e6i)T^{2} \) |
| 17 | \( 1 + (-492. - 2.79e3i)T + (-2.26e7 + 8.25e6i)T^{2} \) |
| 23 | \( 1 + (7.82e3 - 2.84e3i)T + (1.13e8 - 9.51e7i)T^{2} \) |
| 29 | \( 1 + (7.79e3 + 1.37e3i)T + (5.58e8 + 2.03e8i)T^{2} \) |
| 31 | \( 1 + (7.94e3 + 4.58e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 7.72e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-3.89e4 - 4.63e4i)T + (-8.24e8 + 4.67e9i)T^{2} \) |
| 43 | \( 1 + (-7.75e4 - 2.82e4i)T + (4.84e9 + 4.06e9i)T^{2} \) |
| 47 | \( 1 + (-1.68e4 + 9.55e4i)T + (-1.01e10 - 3.68e9i)T^{2} \) |
| 53 | \( 1 + (5.28e4 + 1.45e5i)T + (-1.69e10 + 1.42e10i)T^{2} \) |
| 59 | \( 1 + (-2.85e4 + 5.04e3i)T + (3.96e10 - 1.44e10i)T^{2} \) |
| 61 | \( 1 + (-1.65e5 + 6.00e4i)T + (3.94e10 - 3.31e10i)T^{2} \) |
| 67 | \( 1 + (-4.39e5 - 7.74e4i)T + (8.50e10 + 3.09e10i)T^{2} \) |
| 71 | \( 1 + (8.85e4 - 2.43e5i)T + (-9.81e10 - 8.23e10i)T^{2} \) |
| 73 | \( 1 + (-6.64e4 + 5.57e4i)T + (2.62e10 - 1.49e11i)T^{2} \) |
| 79 | \( 1 + (-1.78e5 - 2.12e5i)T + (-4.22e10 + 2.39e11i)T^{2} \) |
| 83 | \( 1 + (3.76e5 - 6.51e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-5.14e5 + 6.12e5i)T + (-8.62e10 - 4.89e11i)T^{2} \) |
| 97 | \( 1 + (7.87e5 - 1.38e5i)T + (7.82e11 - 2.84e11i)T^{2} \) |
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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
−14.14159282389082187942763747159, −12.54377150514980425880441389796, −11.51996476240085920663431571982, −9.987326078569391914323573584165, −9.373031431959286309996715284204, −8.182569118769711427500634544152, −6.86221557079242546205422215428, −4.52949168792922953632012418642, −4.03860562019067208652397110052, −2.14428710148906815888633450779,
0.57314616715313017060444075238, 2.35187756965931402237689664425, 3.62825990463596041559814023699, 5.81455811459997419726224991634, 7.31985878202535800539210129425, 8.056404561192772649302394046180, 9.103662639682770766338888055785, 10.77502210169342631745439754599, 12.05747518753769192035970689375, 12.85910308142701542641541341226
