| L(s) = 1 | + (−12.2 + 10.2i)2-s + (−46.9 − 17.0i)3-s + (44.4 − 252. i)4-s + (−94.2 − 534. i)5-s + (751. − 273. i)6-s + (−4.67e3 + 8.08e3i)7-s + (2.04e3 + 3.54e3i)8-s + (−1.31e4 − 1.10e4i)9-s + (6.65e3 + 5.58e3i)10-s + (−2.64e4 − 4.57e4i)11-s + (−6.39e3 + 1.10e4i)12-s + (1.23e5 − 4.47e4i)13-s + (−2.59e4 − 1.47e5i)14-s + (−4.71e3 + 2.67e4i)15-s + (−6.15e4 − 2.24e4i)16-s + (−1.31e5 + 1.10e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.334 − 0.121i)3-s + (0.0868 − 0.492i)4-s + (−0.0674 − 0.382i)5-s + (0.236 − 0.0861i)6-s + (−0.735 + 1.27i)7-s + (0.176 + 0.306i)8-s + (−0.668 − 0.561i)9-s + (0.210 + 0.176i)10-s + (−0.544 − 0.942i)11-s + (−0.0890 + 0.154i)12-s + (1.19 − 0.434i)13-s + (−0.180 − 1.02i)14-s + (−0.0240 + 0.136i)15-s + (−0.234 − 0.0855i)16-s + (−0.382 + 0.320i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.909717 + 0.272676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.909717 + 0.272676i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (12.2 - 10.2i)T \) |
| 19 | \( 1 + (-2.33e5 - 5.17e5i)T \) |
| good | 3 | \( 1 + (46.9 + 17.0i)T + (1.50e4 + 1.26e4i)T^{2} \) |
| 5 | \( 1 + (94.2 + 534. i)T + (-1.83e6 + 6.68e5i)T^{2} \) |
| 7 | \( 1 + (4.67e3 - 8.08e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.64e4 + 4.57e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.23e5 + 4.47e4i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (1.31e5 - 1.10e5i)T + (2.05e10 - 1.16e11i)T^{2} \) |
| 23 | \( 1 + (1.09e4 - 6.18e4i)T + (-1.69e12 - 6.16e11i)T^{2} \) |
| 29 | \( 1 + (-4.00e6 - 3.36e6i)T + (2.51e12 + 1.42e13i)T^{2} \) |
| 31 | \( 1 + (1.37e6 - 2.37e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 2.04e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.43e7 - 5.22e6i)T + (2.50e14 + 2.10e14i)T^{2} \) |
| 43 | \( 1 + (3.81e6 + 2.16e7i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (-9.91e6 - 8.31e6i)T + (1.94e14 + 1.10e15i)T^{2} \) |
| 53 | \( 1 + (-1.67e7 + 9.51e7i)T + (-3.10e15 - 1.12e15i)T^{2} \) |
| 59 | \( 1 + (6.54e7 - 5.49e7i)T + (1.50e15 - 8.53e15i)T^{2} \) |
| 61 | \( 1 + (3.15e7 - 1.78e8i)T + (-1.09e16 - 3.99e15i)T^{2} \) |
| 67 | \( 1 + (-1.75e8 - 1.46e8i)T + (4.72e15 + 2.67e16i)T^{2} \) |
| 71 | \( 1 + (4.90e7 + 2.78e8i)T + (-4.30e16 + 1.56e16i)T^{2} \) |
| 73 | \( 1 + (3.44e8 + 1.25e8i)T + (4.50e16 + 3.78e16i)T^{2} \) |
| 79 | \( 1 + (-3.13e8 - 1.14e8i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (-2.32e8 + 4.03e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-3.00e7 + 1.09e7i)T + (2.68e17 - 2.25e17i)T^{2} \) |
| 97 | \( 1 + (-1.84e8 + 1.54e8i)T + (1.32e17 - 7.48e17i)T^{2} \) |
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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
−14.67219732655497081560336432344, −13.11263963504325953611578143320, −11.96745518485030465682745917833, −10.67273711038569982067867050392, −9.015438235635923469670770839486, −8.369805927506440362390576506579, −6.26375710360172428125872511984, −5.60709768419896396514385329909, −3.05168529856544319170587534137, −0.804390802794394182575787116292,
0.67626762962204114369964650677, 2.74331584676742321669565252696, 4.39266843990103432245670931797, 6.53885243276213380970293420326, 7.75705484995314763340660899196, 9.438069746657218280305229015086, 10.63229102523252355865137579970, 11.31284873760298437032511300936, 13.00698702328660765258569316968, 13.90263929425425451615851132348
