| L(s) = 1 | + (8.96 + 12.3i)2-s + (−4.81 − 14.8i)3-s + (−52.1 + 160. i)4-s + (137. + 100. i)5-s + (139. − 192. i)6-s + (−91.3 − 29.6i)7-s + (−1.52e3 + 494. i)8-s + (−196. + 142. i)9-s + 2.59e3i·10-s + (−720. − 1.11e3i)11-s + 2.63e3·12-s + (1.47e3 + 2.02e3i)13-s + (−452. − 1.39e3i)14-s + (820. − 2.52e3i)15-s + (−1.10e4 − 7.99e3i)16-s + (−302. + 416. i)17-s + ⋯ |
| L(s) = 1 | + (1.12 + 1.54i)2-s + (−0.178 − 0.549i)3-s + (−0.815 + 2.50i)4-s + (1.10 + 0.800i)5-s + (0.647 − 0.890i)6-s + (−0.266 − 0.0865i)7-s + (−2.97 + 0.965i)8-s + (−0.269 + 0.195i)9-s + 2.59i·10-s + (−0.541 − 0.840i)11-s + 1.52·12-s + (0.670 + 0.922i)13-s + (−0.165 − 0.508i)14-s + (0.242 − 0.747i)15-s + (−2.68 − 1.95i)16-s + (−0.0616 + 0.0848i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.855142 + 2.69163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.855142 + 2.69163i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.81 + 14.8i)T \) |
| 11 | \( 1 + (720. + 1.11e3i)T \) |
| good | 2 | \( 1 + (-8.96 - 12.3i)T + (-19.7 + 60.8i)T^{2} \) |
| 5 | \( 1 + (-137. - 100. i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (91.3 + 29.6i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (-1.47e3 - 2.02e3i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (302. - 416. i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (-1.01e4 + 3.28e3i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 - 2.21e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.11e4 + 3.62e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (1.02e4 - 7.43e3i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (-9.88e3 + 3.04e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-3.16e4 + 1.02e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 + 2.16e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (1.85e4 + 5.70e4i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (1.32e5 - 9.62e4i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (-4.67e3 + 1.43e4i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (1.92e4 - 2.65e4i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 - 4.01e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (3.69e5 + 2.68e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (2.12e5 + 6.90e4i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (4.07e5 + 5.60e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (6.62e5 - 9.11e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 - 4.04e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.05e6 + 7.62e5i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−15.84976598537970099049670553741, −14.44758460861966444878347288498, −13.67707815913413226747529091099, −13.04837522420275511374756148444, −11.28576333063214341114915024370, −9.023202550121440958604814754157, −7.31265514847806584674571292615, −6.37940326026343916395402359595, −5.36988457393675111669634054458, −3.10941769397676481174208551505,
1.24574012351014796455417020085, 3.05370829713375571743160399418, 4.89966915208097416948420353532, 5.69677210693702994604916050973, 9.328727635746088159148617305725, 10.05241380624775049482043877567, 11.25701676975717158105960301231, 12.76296089147650733604700086203, 13.20550894228739474035725469919, 14.53882087470838384137602559546
