| L(s) = 1 | + (−116. + 202. i)3-s + (2.05e3 − 1.18e3i)5-s + (4.08e3 + 4.86e3i)7-s + (−1.75e4 − 3.03e4i)9-s + (−6.25e4 − 3.61e4i)11-s − 788. i·13-s + 5.54e5i·15-s + (−9.32e4 − 5.38e4i)17-s + (4.28e5 + 7.42e5i)19-s + (−1.46e6 + 2.57e5i)21-s + (1.32e6 − 7.63e5i)23-s + (1.82e6 − 3.16e6i)25-s + 3.59e6·27-s + 4.44e5·29-s + (−4.79e6 + 8.30e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.833 + 1.44i)3-s + (1.46 − 0.847i)5-s + (0.642 + 0.766i)7-s + (−0.890 − 1.54i)9-s + (−1.28 − 0.744i)11-s − 0.00765i·13-s + 2.82i·15-s + (−0.270 − 0.156i)17-s + (0.754 + 1.30i)19-s + (−1.64 + 0.288i)21-s + (0.984 − 0.568i)23-s + (0.935 − 1.62i)25-s + 1.30·27-s + 0.116·29-s + (−0.932 + 1.61i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.610352279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610352279\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-4.08e3 - 4.86e3i)T \) |
| good | 3 | \( 1 + (116. - 202. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-2.05e3 + 1.18e3i)T + (9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (6.25e4 + 3.61e4i)T + (1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 788. iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (9.32e4 + 5.38e4i)T + (5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-4.28e5 - 7.42e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.32e6 + 7.63e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 4.44e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (4.79e6 - 8.30e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-9.54e6 - 1.65e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.87e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 5.52e6iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.36e6 - 4.09e6i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.49e7 + 4.31e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (5.13e7 - 8.89e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.48e8 - 8.57e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-6.43e6 - 3.71e6i)T + (1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 4.93e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.48e8 - 8.55e7i)T + (2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (8.02e7 - 4.63e7i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 6.52e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + (4.65e8 - 2.68e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.06e9iT - 7.60e17T^{2} \) |
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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
−12.10814454802050479887867475499, −10.91880537033754382828486340089, −10.17481724064812369919773311062, −9.266055113415000801433848005714, −8.381708928071732980548085311817, −6.02112383859607170861166205540, −5.32280645702645631045561693043, −4.81398309965870962029009953940, −2.89518430437112711256442922656, −1.25746929880847955557442229373,
0.46342870077180361647152797399, 1.74668421902387373549802744874, 2.52994305133074851126513314809, 5.06876647070856522459040789131, 5.93186832592091829903004981193, 7.16111367083810941731714821065, 7.50537245460226257110619222706, 9.491434961571996007316720780686, 10.76999040226684785821274516494, 11.20612159024804652456120085367
