| L(s) = 1 | + (15.9 − 0.207i)2-s + (114. + 47.6i)3-s + (255. − 6.64i)4-s + (−373. + 154. i)5-s + (1.84e3 + 737. i)6-s + (929. + 929. i)7-s + (4.09e3 − 159. i)8-s + (6.30e3 + 6.30e3i)9-s + (−5.94e3 + 2.55e3i)10-s + (6.38e3 − 2.64e3i)11-s + (2.97e4 + 1.14e4i)12-s + (−4.72e4 − 1.95e4i)13-s + (1.50e4 + 1.46e4i)14-s − 5.03e4·15-s + (6.54e4 − 3.40e3i)16-s + 1.02e5i·17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0129i)2-s + (1.41 + 0.587i)3-s + (0.999 − 0.0259i)4-s + (−0.598 + 0.247i)5-s + (1.42 + 0.569i)6-s + (0.386 + 0.386i)7-s + (0.999 − 0.0389i)8-s + (0.961 + 0.961i)9-s + (−0.594 + 0.255i)10-s + (0.436 − 0.180i)11-s + (1.43 + 0.550i)12-s + (−1.65 − 0.685i)13-s + (0.391 + 0.381i)14-s − 0.994·15-s + (0.998 − 0.0519i)16-s + 1.23i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(4.53549 + 1.53828i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.53549 + 1.53828i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.9 + 0.207i)T \) |
| good | 3 | \( 1 + (-114. - 47.6i)T + (4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (373. - 154. i)T + (2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-929. - 929. i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (-6.38e3 + 2.64e3i)T + (1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (4.72e4 + 1.95e4i)T + (5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 1.02e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-7.33e4 + 1.77e5i)T + (-1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (-1.83e5 + 1.83e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (-9.46e4 + 2.28e5i)T + (-3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 + 3.95e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (1.62e6 - 6.74e5i)T + (2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (2.04e6 + 2.04e6i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (-1.24e6 + 5.14e5i)T + (8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 4.29e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (4.72e6 + 1.14e7i)T + (-4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (-2.55e6 - 6.16e6i)T + (-1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (7.77e6 - 1.87e7i)T + (-1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (-2.65e7 - 1.10e7i)T + (2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (-4.95e6 - 4.95e6i)T + 6.45e14iT^{2} \) |
| 73 | \( 1 + (-1.06e7 - 1.06e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 4.67e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (1.32e7 - 3.20e7i)T + (-1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (6.18e7 - 6.18e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 - 9.66e7T + 7.83e15T^{2} \) |
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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
−15.05466100403243441779674953540, −14.25547977738873298193775190286, −12.92344012680262593061621301506, −11.58188381163181374804241882807, −10.08708474932077686185625497321, −8.464999617297055912920305855664, −7.22190119980608375712158720532, −4.94411783494337199443827093117, −3.55051700773036433027021038497, −2.39774207325416109736464220878,
1.73512564034844173789168390007, 3.23444407271259380277461814272, 4.69111772549138078501787443888, 7.13612263442562159241619691146, 7.83534088400369035480951027980, 9.599820997774311934338044794893, 11.69496173926266132660852044259, 12.57488580891344451396099865657, 14.11070989220130468568435377705, 14.30717024869537136283702184867
