| L(s) = 1 | + (−3.23 − 2.35i)2-s + (−2.20 − 21.0i)3-s + (4.94 + 15.2i)4-s + (−41.2 − 71.4i)5-s + (−42.2 + 73.2i)6-s + (−88.1 − 18.7i)7-s + (19.7 − 60.8i)8-s + (−199. + 42.3i)9-s + (−34.5 + 328. i)10-s + (328. − 364. i)11-s + (308. − 137. i)12-s + (−231. − 102. i)13-s + (241. + 267. i)14-s + (−1.41e3 + 1.02e3i)15-s + (−207. + 150. i)16-s + (1.08e3 + 1.20e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.141 − 1.34i)3-s + (0.154 + 0.475i)4-s + (−0.738 − 1.27i)5-s + (−0.479 + 0.830i)6-s + (−0.679 − 0.144i)7-s + (0.109 − 0.336i)8-s + (−0.819 + 0.174i)9-s + (−0.109 + 1.03i)10-s + (0.818 − 0.909i)11-s + (0.619 − 0.275i)12-s + (−0.379 − 0.168i)13-s + (0.328 + 0.365i)14-s + (−1.61 + 1.17i)15-s + (−0.202 + 0.146i)16-s + (0.909 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.326779 + 0.487809i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.326779 + 0.487809i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 31 | \( 1 + (3.87e3 + 3.68e3i)T \) |
| good | 3 | \( 1 + (2.20 + 21.0i)T + (-237. + 50.5i)T^{2} \) |
| 5 | \( 1 + (41.2 + 71.4i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (88.1 + 18.7i)T + (1.53e4 + 6.83e3i)T^{2} \) |
| 11 | \( 1 + (-328. + 364. i)T + (-1.68e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (231. + 102. i)T + (2.48e5 + 2.75e5i)T^{2} \) |
| 17 | \( 1 + (-1.08e3 - 1.20e3i)T + (-1.48e5 + 1.41e6i)T^{2} \) |
| 19 | \( 1 + (-38.0 + 16.9i)T + (1.65e6 - 1.84e6i)T^{2} \) |
| 23 | \( 1 + (906. - 2.79e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.40e3 - 1.02e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 37 | \( 1 + (-986. + 1.70e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-1.27e3 + 1.21e4i)T + (-1.13e8 - 2.40e7i)T^{2} \) |
| 43 | \( 1 + (-1.81e4 + 8.10e3i)T + (9.83e7 - 1.09e8i)T^{2} \) |
| 47 | \( 1 + (7.82e3 - 5.68e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (3.38e4 - 7.19e3i)T + (3.82e8 - 1.70e8i)T^{2} \) |
| 59 | \( 1 + (1.79e3 + 1.71e4i)T + (-6.99e8 + 1.48e8i)T^{2} \) |
| 61 | \( 1 + 1.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (1.66e4 + 2.88e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-4.81e4 + 1.02e4i)T + (1.64e9 - 7.33e8i)T^{2} \) |
| 73 | \( 1 + (5.69e4 - 6.32e4i)T + (-2.16e8 - 2.06e9i)T^{2} \) |
| 79 | \( 1 + (-5.79e4 - 6.43e4i)T + (-3.21e8 + 3.06e9i)T^{2} \) |
| 83 | \( 1 + (5.37e3 - 5.11e4i)T + (-3.85e9 - 8.18e8i)T^{2} \) |
| 89 | \( 1 + (2.61e4 + 8.05e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (4.60e4 + 1.41e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−12.66683053565963471155412962416, −12.40760675737373319704666915089, −11.28087072870358209798507896997, −9.496604610615929356134715923462, −8.333955975899570758583989359342, −7.45759441152114322940897117827, −5.95535127386882609167030587631, −3.70773502894753267553167719391, −1.41556852588943755448655128106, −0.36444290017962341072480021205,
3.12460025054064753461571986166, 4.56932452633148189333967023917, 6.44722835121397393594035421059, 7.52570382651307623716631011577, 9.349637402532108964758568742304, 10.01704581897165355845986616067, 11.01390440068721997970497868784, 12.14969819394400249361850512908, 14.47809679998371305746183175723, 14.82756152380498973748178332522
