| L(s) = 1 | + (0.318 − 0.267i)2-s + (−0.317 + 1.79i)4-s + (2.08 − 0.757i)5-s + (−0.229 − 1.29i)7-s + (0.795 + 1.37i)8-s + (0.460 − 0.797i)10-s + (−4.90 − 1.78i)11-s + (−0.0138 − 0.0116i)13-s + (−0.419 − 0.352i)14-s + (−2.81 − 1.02i)16-s + (−1.56 + 2.71i)17-s + (−0.208 − 0.361i)19-s + (0.702 + 3.98i)20-s + (−2.03 + 0.741i)22-s + (0.179 − 1.01i)23-s + ⋯ |
| L(s) = 1 | + (0.225 − 0.188i)2-s + (−0.158 + 0.899i)4-s + (0.930 − 0.338i)5-s + (−0.0866 − 0.491i)7-s + (0.281 + 0.486i)8-s + (0.145 − 0.252i)10-s + (−1.47 − 0.537i)11-s + (−0.00383 − 0.00321i)13-s + (−0.112 − 0.0941i)14-s + (−0.703 − 0.256i)16-s + (−0.379 + 0.658i)17-s + (−0.0478 − 0.0829i)19-s + (0.157 + 0.891i)20-s + (−0.434 + 0.157i)22-s + (0.0374 − 0.212i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08822 + 0.0260733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08822 + 0.0260733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.318 + 0.267i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-2.08 + 0.757i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.229 + 1.29i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (4.90 + 1.78i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0138 + 0.0116i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.56 - 2.71i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.208 + 0.361i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.179 + 1.01i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.98 + 5.01i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.647 + 3.67i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.21 - 3.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.81 - 2.36i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-7.80 - 2.84i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.23 - 6.99i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 + (3.47 - 1.26i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.20 - 6.80i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (8.44 + 7.08i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.04 - 5.26i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.273 - 0.473i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.374 + 0.314i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.53 - 2.96i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (1.68 + 2.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.34 + 3.40i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.85352606963554956385066399051, −13.34066146467907994077363197348, −12.53166884602824900377401890746, −11.10160931119706436784515248821, −10.04022828521845496297731492635, −8.648415484422967785165682848853, −7.63238781957835695483506512628, −5.92494874650075946188004769330, −4.46187122974390436386343906454, −2.66954639704832544177155038989,
2.37098045680480032471313580067, 4.95526482619938213764591098792, 5.86301089413555323757065108626, 7.15244069758803295593018271259, 8.957404017612768243317791077477, 10.07056128146914920897890981060, 10.72965541029936158153947640946, 12.44239237987208102715746747390, 13.54459698412801507145374910640, 14.24387988539714490787793734074
