| L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.483 + 2.74i)5-s + (−0.173 − 0.984i)6-s + (2.62 + 0.306i)7-s + (0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (2.13 − 1.78i)10-s + (2.61 + 4.52i)11-s + (−0.500 + 0.866i)12-s + (−0.0246 − 0.139i)13-s + (−1.81 − 1.92i)14-s + (−2.13 + 1.78i)15-s + (−0.939 + 0.342i)16-s + (0.0407 − 0.230i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.442 + 0.371i)3-s + (0.0868 + 0.492i)4-s + (−0.216 + 1.22i)5-s + (−0.0708 − 0.402i)6-s + (0.993 + 0.115i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.674 − 0.565i)10-s + (0.787 + 1.36i)11-s + (−0.144 + 0.249i)12-s + (−0.00684 − 0.0388i)13-s + (−0.485 − 0.514i)14-s + (−0.550 + 0.461i)15-s + (−0.234 + 0.0855i)16-s + (0.00987 − 0.0560i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06680 + 0.907872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06680 + 0.907872i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-2.62 - 0.306i)T \) |
| 19 | \( 1 + (3.73 + 2.25i)T \) |
| good | 5 | \( 1 + (0.483 - 2.74i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-2.61 - 4.52i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0246 + 0.139i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.0407 + 0.230i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (5.78 + 2.10i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.13 - 0.777i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 37 | \( 1 + (1.76 + 3.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.00 - 5.67i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.15 + 4.32i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.521 - 2.95i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.66 - 9.46i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.763 + 4.33i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.54 + 2.38i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.75 - 1.47i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-10.8 - 9.09i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-5.08 - 4.26i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.14 + 2.59i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (5.34 + 9.26i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.79 + 4.86i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (8.36 - 3.04i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.39822729889097224897885530000, −9.795553851385038807928924467333, −8.796767957328503978273039141782, −8.004763753740538429446451728292, −7.18620683956539574734093158806, −6.40714115764816074851297566531, −4.70774022373048142460206259155, −3.98918400939260923117801434686, −2.70317992192195809859901907231, −1.84624498911140241948475867487,
0.838876523065295879366757613634, 1.86745723769106996381127841648, 3.73256322133971582903175941217, 4.73161289954539837535668411097, 5.78286491660469075139925728603, 6.65099472673234911167999924389, 7.990303967193313277350821235076, 8.321331075570208567694806211373, 8.828294248485754795028019600675, 9.834352732063345705433438349571
