| L(s) = 1 | + (−8.28 − 13.6i)2-s + (89.5 + 29.1i)3-s + (−118. + 226. i)4-s + (−624. + 19.7i)5-s + (−343. − 1.46e3i)6-s − 1.00e3i·7-s + (4.08e3 − 254. i)8-s + (1.87e3 + 1.35e3i)9-s + (5.44e3 + 8.38e3i)10-s + (5.70e3 + 7.85e3i)11-s + (−1.72e4 + 1.68e4i)12-s + (−3.17e4 − 2.30e4i)13-s + (−1.37e4 + 8.31e3i)14-s + (−5.65e4 − 1.64e4i)15-s + (−3.73e4 − 5.38e4i)16-s + (1.74e4 + 5.36e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.517 − 0.855i)2-s + (1.10 + 0.359i)3-s + (−0.463 + 0.886i)4-s + (−0.999 + 0.0315i)5-s + (−0.265 − 1.13i)6-s − 0.418i·7-s + (0.998 − 0.0622i)8-s + (0.285 + 0.207i)9-s + (0.544 + 0.838i)10-s + (0.389 + 0.536i)11-s + (−0.831 + 0.813i)12-s + (−1.11 − 0.806i)13-s + (−0.357 + 0.216i)14-s + (−1.11 − 0.324i)15-s + (−0.570 − 0.821i)16-s + (0.208 + 0.641i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0208i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 - 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.59175 + 0.0165755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59175 + 0.0165755i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8.28 + 13.6i)T \) |
| 5 | \( 1 + (624. - 19.7i)T \) |
| good | 3 | \( 1 + (-89.5 - 29.1i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 1.00e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-5.70e3 - 7.85e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (3.17e4 + 2.30e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-1.74e4 - 5.36e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-1.68e5 + 5.46e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-4.83e4 - 6.65e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (1.57e5 - 4.83e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (7.53e5 - 2.44e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-3.00e6 - 2.18e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-2.51e6 - 1.83e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 5.08e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-4.55e6 - 1.48e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-1.79e6 + 5.51e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-1.32e6 + 1.82e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (1.41e7 - 1.03e7i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-2.79e7 + 9.08e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-7.40e6 - 2.40e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-7.52e6 + 5.46e6i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (2.50e7 + 8.15e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-3.74e7 + 1.21e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-3.63e7 + 2.64e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (1.64e7 - 5.05e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.17489998668956460449107646883, −11.14905766990044596266057744452, −9.936773451843103949211799621361, −9.181021100885896625887088440079, −7.954418280216120254103846708356, −7.40465531686452822840603011280, −4.65710527262614037834147591124, −3.55766533612505408642791088381, −2.72198139715730510018642111736, −0.943072038205508235761772537385,
0.62301877899471619156223224603, 2.36971724681336852259341366029, 3.96394226595650099370911717359, 5.50330993429860195575326002400, 7.24044162544634379706634271141, 7.69608089220328659956453731946, 8.880748592520488241334272107738, 9.470941660799419174266499700204, 11.19667474715213043500410740608, 12.31223958282302037424885851284
