| L(s) = 1 | + (−1.03 − 1.50i)2-s + (−0.00317 + 0.00365i)3-s + (−0.484 + 1.25i)4-s + (−2.03 + 2.82i)5-s + (0.00879 + 0.00102i)6-s + (3.18 − 1.87i)7-s + (−1.17 + 0.275i)8-s + (0.414 + 2.97i)9-s + (6.36 + 0.147i)10-s + (2.99 − 3.14i)11-s + (−0.00302 − 0.00574i)12-s + (3.07 − 4.96i)13-s + (−6.11 − 2.87i)14-s + (−0.00386 − 0.0164i)15-s + (3.60 + 3.29i)16-s + (−2.16 − 3.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.730 − 1.06i)2-s + (−0.00183 + 0.00210i)3-s + (−0.242 + 0.625i)4-s + (−0.909 + 1.26i)5-s + (0.00359 + 0.000416i)6-s + (1.20 − 0.708i)7-s + (−0.414 + 0.0974i)8-s + (0.138 + 0.990i)9-s + (2.01 + 0.0465i)10-s + (0.904 − 0.946i)11-s + (−0.000874 − 0.00165i)12-s + (0.852 − 1.37i)13-s + (−1.63 − 0.767i)14-s + (−0.000997 − 0.00424i)15-s + (0.902 + 0.822i)16-s + (−0.525 − 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.744026 - 0.514173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744026 - 0.514173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (2.16 + 3.50i)T \) |
| good | 2 | \( 1 + (1.03 + 1.50i)T + (-0.722 + 1.86i)T^{2} \) |
| 3 | \( 1 + (0.00317 - 0.00365i)T + (-0.414 - 2.97i)T^{2} \) |
| 5 | \( 1 + (2.03 - 2.82i)T + (-1.58 - 4.74i)T^{2} \) |
| 7 | \( 1 + (-3.18 + 1.87i)T + (3.40 - 6.11i)T^{2} \) |
| 11 | \( 1 + (-2.99 + 3.14i)T + (-0.508 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.07 + 4.96i)T + (-5.79 - 11.6i)T^{2} \) |
| 19 | \( 1 + (-3.71 - 2.54i)T + (6.86 + 17.7i)T^{2} \) |
| 23 | \( 1 + (-2.54 - 4.32i)T + (-11.1 + 20.0i)T^{2} \) |
| 29 | \( 1 + (-5.28 + 0.122i)T + (28.9 - 1.33i)T^{2} \) |
| 31 | \( 1 + (-0.244 + 1.49i)T + (-29.3 - 9.85i)T^{2} \) |
| 37 | \( 1 + (8.58 - 2.65i)T + (30.5 - 20.9i)T^{2} \) |
| 41 | \( 1 + (0.0637 + 0.918i)T + (-40.6 + 5.66i)T^{2} \) |
| 43 | \( 1 + (-0.111 + 2.40i)T + (-42.8 - 3.96i)T^{2} \) |
| 47 | \( 1 + (-6.56 + 4.96i)T + (12.8 - 45.2i)T^{2} \) |
| 53 | \( 1 + (-3.47 + 0.484i)T + (50.9 - 14.5i)T^{2} \) |
| 59 | \( 1 + (-3.14 - 5.65i)T + (-31.0 + 50.1i)T^{2} \) |
| 61 | \( 1 + (1.04 - 9.01i)T + (-59.3 - 13.9i)T^{2} \) |
| 67 | \( 1 + (2.18 - 0.407i)T + (62.4 - 24.2i)T^{2} \) |
| 71 | \( 1 + (8.06 + 2.09i)T + (62.0 + 34.5i)T^{2} \) |
| 73 | \( 1 + (-7.73 - 2.79i)T + (56.1 + 46.6i)T^{2} \) |
| 79 | \( 1 + (-0.336 + 1.59i)T + (-72.2 - 31.9i)T^{2} \) |
| 83 | \( 1 + (10.6 - 3.56i)T + (66.2 - 50.0i)T^{2} \) |
| 89 | \( 1 + (2.88 + 4.66i)T + (-39.6 + 79.6i)T^{2} \) |
| 97 | \( 1 + (-0.415 + 1.59i)T + (-84.7 - 47.2i)T^{2} \) |
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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
−11.45508495162843319702481728553, −10.66142219039362220154261786283, −10.35448799243941202938818768301, −8.712052551324004986045759228821, −7.954327920374056785473245232488, −7.11458929613528827558931746171, −5.51917039768044042816029368174, −3.82393988161553450469378319028, −2.91374494812924537792461722512, −1.15413675383463781497999697077,
1.32058490096323966387916486774, 4.03004600271860960008156497168, 4.91023165753391526955398715224, 6.37251318923124040008510610594, 7.19131023295467614357094956087, 8.491541435371539148250511392472, 8.743859205971788902864704922853, 9.418969523470948421383518909339, 11.30141382044146537588127691739, 12.10151615933675092789626896325
