| L(s) = 1 | + (−5.21 − 0.456i)2-s + (1.63 − 4.93i)3-s + (19.1 + 3.37i)4-s + (−2.66 + 10.8i)5-s + (−10.8 + 24.9i)6-s + (7.48 + 5.23i)7-s + (−57.8 − 15.5i)8-s + (−21.6 − 16.1i)9-s + (18.8 − 55.4i)10-s + (−8.10 + 22.2i)11-s + (48.0 − 88.8i)12-s + (−5.04 − 57.6i)13-s + (−36.6 − 30.7i)14-s + (49.1 + 30.9i)15-s + (148. + 54.1i)16-s + (−10.4 + 2.78i)17-s + ⋯ |
| L(s) = 1 | + (−1.84 − 0.161i)2-s + (0.315 − 0.948i)3-s + (2.39 + 0.421i)4-s + (−0.238 + 0.971i)5-s + (−0.735 + 1.69i)6-s + (0.404 + 0.282i)7-s + (−2.55 − 0.685i)8-s + (−0.800 − 0.598i)9-s + (0.596 − 1.75i)10-s + (−0.222 + 0.610i)11-s + (1.15 − 2.13i)12-s + (−0.107 − 1.23i)13-s + (−0.699 − 0.587i)14-s + (0.846 + 0.532i)15-s + (2.32 + 0.845i)16-s + (−0.148 + 0.0397i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0163908 + 0.0485111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0163908 + 0.0485111i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.63 + 4.93i)T \) |
| 5 | \( 1 + (2.66 - 10.8i)T \) |
| good | 2 | \( 1 + (5.21 + 0.456i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-7.48 - 5.23i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (8.10 - 22.2i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (5.04 + 57.6i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (10.4 - 2.78i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (73.3 - 42.3i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (107. + 153. i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (104. - 87.3i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (49.2 - 279. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-34.4 - 128. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-28.3 + 33.7i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-2.04 - 4.38i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (46.5 - 66.5i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (473. - 473. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-808. + 294. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-19.2 - 109. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (725. - 63.4i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (974. + 562. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (122. - 455. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (59.4 + 70.8i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-97.8 + 1.11e3i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-112. - 194. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.04e3 - 489. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.71441598464659692401806358226, −11.91837240809331908386249737323, −10.77509780903540479289811576809, −10.15896929751937586709657491002, −8.681267113006445319189419406114, −7.988002503174791194432581680152, −7.16769618723423000686273242887, −6.18297745997347754643084705337, −2.93619357588852823231955512223, −1.84835305905154041798302329773,
0.04171615884012669976192297881, 1.96917871497298948584612210805, 4.16736100136138732655306673639, 5.84765188311137321603431922231, 7.56821683357471590536055683149, 8.387004258803918481831317476107, 9.211595726370523296923898255457, 9.841227945020323769228291556363, 11.19375172992098196938974582384, 11.55591507393907955002602826898
