| L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.818 − 1.52i)3-s + (0.173 + 0.984i)4-s + (1.25 + 0.221i)5-s + (−1.60 + 0.643i)6-s + (−1.62 − 2.08i)7-s + (0.500 − 0.866i)8-s + (−1.66 − 2.49i)9-s + (−0.819 − 0.977i)10-s + (−4.03 + 2.32i)11-s + (1.64 + 0.540i)12-s + (−2.55 + 0.451i)13-s + (−0.0952 + 2.64i)14-s + (1.36 − 1.73i)15-s + (−0.939 + 0.342i)16-s + (−4.31 − 0.761i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.472 − 0.881i)3-s + (0.0868 + 0.492i)4-s + (0.561 + 0.0990i)5-s + (−0.656 + 0.262i)6-s + (−0.614 − 0.788i)7-s + (0.176 − 0.306i)8-s + (−0.553 − 0.832i)9-s + (−0.259 − 0.308i)10-s + (−1.21 + 0.702i)11-s + (0.475 + 0.156i)12-s + (−0.709 + 0.125i)13-s + (−0.0254 + 0.706i)14-s + (0.352 − 0.448i)15-s + (−0.234 + 0.0855i)16-s + (−1.04 − 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112556 + 0.585807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112556 + 0.585807i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.818 + 1.52i)T \) |
| 7 | \( 1 + (1.62 + 2.08i)T \) |
| 19 | \( 1 + (-3.56 + 2.51i)T \) |
| good | 5 | \( 1 + (-1.25 - 0.221i)T + (4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (4.03 - 2.32i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.55 - 0.451i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (4.31 + 0.761i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.39 - 3.83i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (2.24 + 0.816i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + 3.38iT - 31T^{2} \) |
| 37 | \( 1 + (-3.51 + 2.02i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.67 + 9.51i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.447 - 0.375i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.32 + 0.938i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (1.35 + 7.66i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.221 + 1.25i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (9.72 + 3.54i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-5.67 - 6.76i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0380 - 0.0319i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.66 - 2.23i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.39 - 3.83i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.668 - 0.385i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.22 + 3.54i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.07 - 5.68i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−9.628129144796139012548634455143, −9.245708274033218399306679151102, −7.87382127400783834712076152238, −7.39775135747729482712148824687, −6.66114758625893439035903476123, −5.44088795054888308031785920989, −4.00722953658785207276372850432, −2.71421216426128299342847864823, −2.02170806429192349246661864290, −0.30067176613274645259299388728,
2.26864167682292537461273825561, 3.10140120865028393711171937261, 4.66808236916453614536907242755, 5.56289491393445754161452358319, 6.18328479175354371144119407168, 7.59878624936493580659653698263, 8.339615495627783201312695299387, 9.150822860250409035040869126083, 9.727677626752308341641514768364, 10.42710230891507358695977712587
