| L(s) = 1 | + (−1.54 − 1.16i)2-s + (0.0205 + 0.444i)3-s + (0.483 + 1.70i)4-s + (0.215 + 0.0953i)5-s + (0.488 − 0.712i)6-s + (−0.568 − 0.472i)7-s + (−0.162 + 0.420i)8-s + (2.78 − 0.258i)9-s + (−0.222 − 0.400i)10-s + (1.33 + 0.741i)11-s + (−0.746 + 0.250i)12-s + (−1.70 − 0.662i)13-s + (0.328 + 1.39i)14-s + (−0.0379 + 0.0979i)15-s + (3.74 − 2.32i)16-s + (3.80 + 1.59i)17-s + ⋯ |
| L(s) = 1 | + (−1.09 − 0.827i)2-s + (0.0118 + 0.256i)3-s + (0.241 + 0.850i)4-s + (0.0965 + 0.0426i)5-s + (0.199 − 0.291i)6-s + (−0.215 − 0.178i)7-s + (−0.0575 + 0.148i)8-s + (0.929 − 0.0861i)9-s + (−0.0705 − 0.126i)10-s + (0.401 + 0.223i)11-s + (−0.215 + 0.0722i)12-s + (−0.474 − 0.183i)13-s + (0.0878 + 0.373i)14-s + (−0.00980 + 0.0253i)15-s + (0.937 − 0.580i)16-s + (0.922 + 0.386i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.706496 - 0.363991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.706496 - 0.363991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-3.80 - 1.59i)T \) |
| good | 2 | \( 1 + (1.54 + 1.16i)T + (0.547 + 1.92i)T^{2} \) |
| 3 | \( 1 + (-0.0205 - 0.444i)T + (-2.98 + 0.276i)T^{2} \) |
| 5 | \( 1 + (-0.215 - 0.0953i)T + (3.36 + 3.69i)T^{2} \) |
| 7 | \( 1 + (0.568 + 0.472i)T + (1.28 + 6.88i)T^{2} \) |
| 11 | \( 1 + (-1.33 - 0.741i)T + (5.79 + 9.35i)T^{2} \) |
| 13 | \( 1 + (1.70 + 0.662i)T + (9.60 + 8.75i)T^{2} \) |
| 19 | \( 1 + (-4.08 + 3.08i)T + (5.19 - 18.2i)T^{2} \) |
| 23 | \( 1 + (-3.39 - 2.81i)T + (4.22 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.33 + 7.78i)T + (-15.2 - 24.6i)T^{2} \) |
| 31 | \( 1 + (1.00 + 2.28i)T + (-20.8 + 22.9i)T^{2} \) |
| 37 | \( 1 + (0.0591 - 0.176i)T + (-29.5 - 22.2i)T^{2} \) |
| 41 | \( 1 + (-2.73 + 0.126i)T + (40.8 - 3.78i)T^{2} \) |
| 43 | \( 1 + (-2.84 + 4.59i)T + (-19.1 - 38.4i)T^{2} \) |
| 47 | \( 1 + (1.11 - 12.0i)T + (-46.1 - 8.63i)T^{2} \) |
| 53 | \( 1 + (-4.89 + 0.453i)T + (52.0 - 9.73i)T^{2} \) |
| 59 | \( 1 + (0.696 - 3.72i)T + (-55.0 - 21.3i)T^{2} \) |
| 61 | \( 1 + (6.73 - 9.83i)T + (-22.0 - 56.8i)T^{2} \) |
| 67 | \( 1 + (-6.80 - 9.01i)T + (-18.3 + 64.4i)T^{2} \) |
| 71 | \( 1 + (9.67 - 11.6i)T + (-13.0 - 69.7i)T^{2} \) |
| 73 | \( 1 + (-6.54 - 1.53i)T + (65.3 + 32.5i)T^{2} \) |
| 79 | \( 1 + (-2.03 - 0.283i)T + (75.9 + 21.6i)T^{2} \) |
| 83 | \( 1 + (3.89 + 4.26i)T + (-7.65 + 82.6i)T^{2} \) |
| 89 | \( 1 + (6.44 - 2.49i)T + (65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (-4.75 + 5.73i)T + (-17.8 - 95.3i)T^{2} \) |
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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
−11.53670193790965140068198184480, −10.40056839943050041055537039820, −9.828011145623493169708872925702, −9.263553772198011815862899670369, −7.983080370550022649201911557207, −7.13076952428162618819168797879, −5.61309254838149830881139581133, −4.15503748692268004381486876897, −2.71712734001311216424919347395, −1.13003328492254443279895337176,
1.27416631902297170276319808724, 3.44048891745466961866104053083, 5.16044360426330792187864359207, 6.43499473248368394131155186655, 7.24874988657769330977304070919, 7.957869637049153183457192808282, 9.146988778014393370945526314797, 9.715632067773259669730554559704, 10.62983339368832277852135325729, 12.09292203486001938060873047706
