| L(s) = 1 | + (1.89 + 0.756i)2-s + (−1.72 − 0.187i)3-s + (0.132 + 0.125i)4-s + (3.62 − 4.27i)5-s + (−3.12 − 1.65i)6-s + (−2.21 − 1.33i)7-s + (−3.27 − 7.08i)8-s + (2.92 + 0.644i)9-s + (10.1 − 5.37i)10-s + (14.9 + 0.812i)11-s + (−0.204 − 0.240i)12-s + (−3.95 − 17.9i)13-s + (−3.20 − 4.21i)14-s + (−7.04 + 6.67i)15-s + (−0.903 − 16.6i)16-s + (11.3 − 6.85i)17-s + ⋯ |
| L(s) = 1 | + (0.949 + 0.378i)2-s + (−0.573 − 0.0624i)3-s + (0.0330 + 0.0313i)4-s + (0.725 − 0.854i)5-s + (−0.521 − 0.276i)6-s + (−0.316 − 0.190i)7-s + (−0.409 − 0.885i)8-s + (0.325 + 0.0716i)9-s + (1.01 − 0.537i)10-s + (1.36 + 0.0738i)11-s + (−0.0170 − 0.0200i)12-s + (−0.304 − 1.38i)13-s + (−0.228 − 0.301i)14-s + (−0.469 + 0.445i)15-s + (−0.0564 − 1.04i)16-s + (0.669 − 0.403i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.83317 - 0.822518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83317 - 0.822518i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.187i)T \) |
| 59 | \( 1 + (-35.4 + 47.1i)T \) |
| good | 2 | \( 1 + (-1.89 - 0.756i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (-3.62 + 4.27i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (2.21 + 1.33i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (-14.9 - 0.812i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (3.95 + 17.9i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (-11.3 + 6.85i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (7.70 - 27.7i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (12.2 - 8.30i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (4.15 + 10.4i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (-47.1 + 13.0i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (14.9 - 32.3i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (20.4 - 30.0i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (-13.2 + 0.717i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (34.0 - 28.9i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (20.9 - 39.5i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (-68.9 - 27.4i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (16.4 + 35.5i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (-77.8 - 91.6i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (-17.4 - 22.9i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (-143. + 15.6i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (-2.37 - 7.03i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (37.0 - 14.7i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (-84.0 + 110. i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.52093575254104415601011385404, −11.83076808561057078027589780416, −10.01263773151255984561087758416, −9.681439525456019794082544573628, −8.104735909735240319755792049781, −6.52532394531193870683934941011, −5.79602020384145621753988196708, −4.91146251323460407129376052146, −3.65353724149194851808396867752, −1.07861578952963803265629073872,
2.22179063425275054047198177548, 3.70538597450014230382959795637, 4.84787121407544970301366655195, 6.24364856326685105646815771730, 6.77395392821453591857676121049, 8.777457214864527825862890639911, 9.736553277159473990432909762752, 10.93691577415334706449372322648, 11.79052638100275619534304982023, 12.44432282272937748455668917888
