| L(s) = 1 | + (−5.21 − 0.456i)2-s + (−2.75 + 4.40i)3-s + (19.1 + 3.37i)4-s + (11.1 + 0.572i)5-s + (16.3 − 21.7i)6-s + (26.1 + 18.3i)7-s + (−57.9 − 15.5i)8-s + (−11.8 − 24.2i)9-s + (−58.0 − 8.08i)10-s + (15.2 − 41.8i)11-s + (−67.6 + 75.1i)12-s + (−1.53 − 17.5i)13-s + (−128. − 107. i)14-s + (−33.2 + 47.6i)15-s + (149. + 54.3i)16-s + (47.5 − 12.7i)17-s + ⋯ |
| L(s) = 1 | + (−1.84 − 0.161i)2-s + (−0.529 + 0.848i)3-s + (2.39 + 0.422i)4-s + (0.998 + 0.0511i)5-s + (1.11 − 1.47i)6-s + (1.41 + 0.989i)7-s + (−2.56 − 0.686i)8-s + (−0.438 − 0.898i)9-s + (−1.83 − 0.255i)10-s + (0.417 − 1.14i)11-s + (−1.62 + 1.80i)12-s + (−0.0327 − 0.374i)13-s + (−2.44 − 2.05i)14-s + (−0.572 + 0.819i)15-s + (2.33 + 0.848i)16-s + (0.678 − 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.841204 + 0.168670i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.841204 + 0.168670i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.75 - 4.40i)T \) |
| 5 | \( 1 + (-11.1 - 0.572i)T \) |
| good | 2 | \( 1 + (5.21 + 0.456i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-26.1 - 18.3i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (-15.2 + 41.8i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (1.53 + 17.5i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (-47.5 + 12.7i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (-79.6 + 46.0i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (51.9 + 74.2i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (-29.8 + 25.0i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-21.9 + 124. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (19.5 + 72.9i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (87.5 - 104. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-37.3 - 80.0i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (8.26 - 11.7i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (427. - 427. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-370. + 134. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-66.3 - 376. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (491. - 42.9i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (-921. - 531. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-242. + 906. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (452. + 539. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (60.1 - 687. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-560. - 970. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-371. + 173. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.06286532756386904398412660320, −11.35536242646512356669839539001, −10.63157198645080570473988641408, −9.589154506311762409258530404006, −8.897556691389896571155884717677, −7.998691325485148554116000092874, −6.25582497829382476615533008332, −5.33415151284460889619038490336, −2.71747491750032170711101028463, −1.07404357439039605702342644118,
1.26630556528158078753168030519, 1.81703237826646897516652524275, 5.25921714767933167108355042525, 6.70078300001392999756529671283, 7.45499111972608627508274509263, 8.283432500383826273381880694014, 9.683465095139247711980205961453, 10.40723025767863150073298848034, 11.38145924444151415866391562036, 12.23952988698905626529464443348
