| L(s) = 1 | + (−1.30 + 0.549i)2-s + (−1.04 + 1.38i)3-s + (1.39 − 1.43i)4-s + (0.437 + 0.159i)5-s + (0.595 − 2.37i)6-s + (3.46 + 0.610i)7-s + (−1.03 + 2.63i)8-s + (−0.832 − 2.88i)9-s + (−0.657 + 0.0329i)10-s + (0.485 + 1.33i)11-s + (0.529 + 3.42i)12-s + (1.62 + 1.93i)13-s + (−4.84 + 1.10i)14-s + (−0.675 + 0.439i)15-s + (−0.103 − 3.99i)16-s + (−0.667 + 0.385i)17-s + ⋯ |
| L(s) = 1 | + (−0.921 + 0.388i)2-s + (−0.601 + 0.799i)3-s + (0.697 − 0.716i)4-s + (0.195 + 0.0711i)5-s + (0.243 − 0.969i)6-s + (1.30 + 0.230i)7-s + (−0.364 + 0.931i)8-s + (−0.277 − 0.960i)9-s + (−0.207 + 0.0104i)10-s + (0.146 + 0.402i)11-s + (0.152 + 0.988i)12-s + (0.449 + 0.535i)13-s + (−1.29 + 0.295i)14-s + (−0.174 + 0.113i)15-s + (−0.0258 − 0.999i)16-s + (−0.161 + 0.0934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0775 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0775 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.513234 + 0.554702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.513234 + 0.554702i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 0.549i)T \) |
| 3 | \( 1 + (1.04 - 1.38i)T \) |
| good | 5 | \( 1 + (-0.437 - 0.159i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-3.46 - 0.610i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.485 - 1.33i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.62 - 1.93i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.667 - 0.385i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.66 - 6.34i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.25 - 7.12i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.15 + 2.64i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-7.80 + 1.37i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.79 + 1.61i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.74 - 2.07i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.99 - 0.726i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.82 + 10.3i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 8.73T + 53T^{2} \) |
| 59 | \( 1 + (-0.818 + 2.24i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (3.02 + 0.533i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.04 - 3.39i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.70 + 9.88i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.96 + 5.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.94 - 10.6i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.41 + 8.83i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-8.59 - 4.96i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.95 - 0.710i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.92571864550750852014687463654, −11.48505507649534996667696259720, −10.48397040506127652773628547581, −9.717963625844133295048175611460, −8.679610012449281038883570037725, −7.77022541307712594985999045922, −6.33274787124569690897075740885, −5.48205909230621483199162180133, −4.19964630262095114492099372673, −1.78563863336526154881539729924,
1.02122503609869474891670696693, 2.47139197041929280646991126520, 4.63406309033980536628564761059, 6.12422098801447721445632358991, 7.18484980762715815542398064798, 8.170125489733175990220167806042, 8.859724648403660844722736322441, 10.50481437337474076470415008277, 11.06289822233755984858786724375, 11.75216138454731860764795419313
