| L(s) = 1 | + (8 − 13.8i)2-s + (3 + 5.19i)3-s + (−127. − 221. i)4-s + (−280 + 484. i)5-s + 96·6-s − 4.09e3·8-s + (9.82e3 − 1.70e4i)9-s + (4.47e3 + 7.75e3i)10-s + (2.70e4 + 4.68e4i)11-s + (768. − 1.33e3i)12-s − 1.13e5·13-s − 3.36e3·15-s + (−3.27e4 + 5.67e4i)16-s + (−3.13e3 − 5.42e3i)17-s + (−1.57e5 − 2.72e5i)18-s + (−1.28e5 + 2.22e5i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.0213 + 0.0370i)3-s + (−0.249 − 0.433i)4-s + (−0.200 + 0.347i)5-s + 0.0302·6-s − 0.353·8-s + (0.499 − 0.864i)9-s + (0.141 + 0.245i)10-s + (0.557 + 0.965i)11-s + (0.0106 − 0.0185i)12-s − 1.09·13-s − 0.0171·15-s + (−0.125 + 0.216i)16-s + (−0.00909 − 0.0157i)17-s + (−0.352 − 0.611i)18-s + (−0.226 + 0.391i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.25324 - 0.142991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25324 - 0.142991i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 + 13.8i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-3 - 5.19i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (280 - 484. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-2.70e4 - 4.68e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.13e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (3.13e3 + 5.42e3i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.28e5 - 2.22e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.33e5 + 2.30e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 1.57e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.31e6 - 4.01e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-5.97e6 + 1.03e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 2.19e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.75e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (2.64e7 - 4.58e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (8.11e6 + 1.40e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-7.02e7 - 1.21e8i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.01e8 + 1.75e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (7.68e7 + 1.33e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.79e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-2.02e8 - 3.49e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-6.53e7 + 1.13e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 4.20e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.34e8 + 4.06e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 8.72e8T + 7.60e17T^{2} \) |
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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
−12.31377659562315199167610958342, −11.11163265956114710154072667396, −9.938857638964216529498951685198, −9.221423674953711280607329796132, −7.46595616542399331706859180227, −6.44109128350724181811848065330, −4.81020535206744962485018822472, −3.77590167184934447776544958662, −2.41026522082941865530767759208, −0.985137403339233008378496213823,
0.67138322272956082428821931454, 2.52847649096980083939610432545, 4.16723520551407997653166404845, 5.13887088922836023432701917768, 6.48427084751048912876400324667, 7.65330778295206546852670825937, 8.589207119582646278380693341788, 9.856587672622495908841259744738, 11.21032702198106835410692582258, 12.31648491966215198883299541082
