| L(s) = 1 | + (−2.18 − 1.52i)2-s + (−1.19 − 1.25i)3-s + (1.73 + 4.77i)4-s + (1.64 + 1.51i)5-s + (0.674 + 4.56i)6-s + (1.30 + 2.78i)7-s + (2.12 − 7.92i)8-s + (−0.166 + 2.99i)9-s + (−1.28 − 5.81i)10-s + (−0.426 − 0.508i)11-s + (3.94 − 7.87i)12-s + (−0.269 − 0.384i)13-s + (1.42 − 8.06i)14-s + (−0.0619 − 3.87i)15-s + (−8.94 + 7.50i)16-s + (1.20 + 4.50i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 − 1.07i)2-s + (−0.687 − 0.726i)3-s + (0.869 + 2.38i)4-s + (0.737 + 0.675i)5-s + (0.275 + 1.86i)6-s + (0.491 + 1.05i)7-s + (0.750 − 2.80i)8-s + (−0.0555 + 0.998i)9-s + (−0.407 − 1.83i)10-s + (−0.128 − 0.153i)11-s + (1.13 − 2.27i)12-s + (−0.0746 − 0.106i)13-s + (0.380 − 2.15i)14-s + (−0.0159 − 0.999i)15-s + (−2.23 + 1.87i)16-s + (0.292 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.485306 - 0.117482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.485306 - 0.117482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.19 + 1.25i)T \) |
| 5 | \( 1 + (-1.64 - 1.51i)T \) |
| good | 2 | \( 1 + (2.18 + 1.52i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 2.78i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (0.426 + 0.508i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.269 + 0.384i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.20 - 4.50i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.80 + 2.77i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0602 + 0.0280i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.434 - 2.46i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.76 + 0.642i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (2.44 - 0.656i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.62 - 0.286i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.732 + 8.36i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (3.71 - 1.73i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-7.52 - 7.52i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.49 + 2.93i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (5.84 + 2.12i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (4.48 - 3.13i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (5.33 + 3.07i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.6 - 3.64i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.34 - 0.941i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.25 + 4.64i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (3.08 + 5.33i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.5 + 1.18i)T + (95.5 + 16.8i)T^{2} \) |
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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
−12.60359425490718266284001983648, −11.85532293022769901968422614964, −11.00878459780845801932433225295, −10.26759978200724029029828414983, −9.105226263762607691882642450407, −8.072139695543871389558829291808, −6.99664187591953435243661182291, −5.63423498618656484720424051503, −2.81638255426190371145795327954, −1.65263123006231444481243593591,
1.03600205136135443368543154237, 4.80085663284980986562493374842, 5.73899874190580611601979988162, 6.96486046281736828760350509972, 8.042401127515532634939074543981, 9.354234458069961258708604741988, 9.882475277607882405341630794264, 10.71821796997650454625459140712, 11.85047354931462533711785476497, 13.75684232060402899222487228547
