| L(s) = 1 | + (0.766 − 0.642i)2-s + (2.36 + 0.861i)3-s + (0.173 − 0.984i)4-s + (−0.427 − 2.42i)5-s + (2.36 − 0.861i)6-s + (1.39 − 2.41i)7-s + (−0.500 − 0.866i)8-s + (2.56 + 2.15i)9-s + (−1.88 − 1.58i)10-s + (0.839 + 1.45i)11-s + (1.26 − 2.18i)12-s + (−5.96 + 2.16i)13-s + (−0.485 − 2.75i)14-s + (1.07 − 6.10i)15-s + (−0.939 − 0.342i)16-s + (3.80 − 3.19i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (1.36 + 0.497i)3-s + (0.0868 − 0.492i)4-s + (−0.191 − 1.08i)5-s + (0.966 − 0.351i)6-s + (0.527 − 0.914i)7-s + (−0.176 − 0.306i)8-s + (0.855 + 0.718i)9-s + (−0.595 − 0.499i)10-s + (0.253 + 0.438i)11-s + (0.363 − 0.630i)12-s + (−1.65 + 0.601i)13-s + (−0.129 − 0.735i)14-s + (0.277 − 1.57i)15-s + (−0.234 − 0.0855i)16-s + (0.923 − 0.774i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.58478 - 1.66657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.58478 - 1.66657i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-2.36 - 0.861i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.427 + 2.42i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.39 + 2.41i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.839 - 1.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.96 - 2.16i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.80 + 3.19i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.433 - 2.46i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.54 - 3.81i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (3.64 - 6.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.550T + 37T^{2} \) |
| 41 | \( 1 + (-2.45 - 0.891i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.498 - 2.82i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.571 - 0.479i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.255 - 1.44i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (3.80 - 3.18i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.62 + 9.19i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.85 - 7.43i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.21 + 6.89i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.81 - 2.11i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-5.55 - 2.02i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (7.58 - 13.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.48 - 2.36i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-11.0 + 9.24i)T + (16.8 - 95.5i)T^{2} \) |
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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
−9.920441131738134827855452752304, −9.561898920168292808234904786941, −8.687924769676569822598481754606, −7.72447078530319570555659562483, −7.04892375369679519248334814534, −5.09081271061632637732477565185, −4.64422119272475722871044404570, −3.77100832284335538789284878623, −2.64694916017560313305754524145, −1.33611719783546838436684404577,
2.26858498608402045289297983136, 2.82195561389613992426968025912, 3.82522406114794041168132624697, 5.23922482921714779827945236166, 6.24480559507708170995375893054, 7.31029106618545368745324522157, 7.85219766687992666791084122049, 8.508909041918931205339556449302, 9.523832781996714451516332739446, 10.49850675940079780732371827451
