| L(s) = 1 | + (0.766 − 0.642i)2-s + (−2.62 − 0.955i)3-s + (0.173 − 0.984i)4-s + (−0.407 − 2.30i)5-s + (−2.62 + 0.955i)6-s + (0.642 − 1.11i)7-s + (−0.500 − 0.866i)8-s + (3.68 + 3.08i)9-s + (−1.79 − 1.50i)10-s + (−2.87 − 4.98i)11-s + (−1.39 + 2.41i)12-s + (−0.285 + 0.104i)13-s + (−0.222 − 1.26i)14-s + (−1.13 + 6.45i)15-s + (−0.939 − 0.342i)16-s + (3.20 − 2.69i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−1.51 − 0.551i)3-s + (0.0868 − 0.492i)4-s + (−0.182 − 1.03i)5-s + (−1.07 + 0.390i)6-s + (0.242 − 0.420i)7-s + (−0.176 − 0.306i)8-s + (1.22 + 1.02i)9-s + (−0.567 − 0.476i)10-s + (−0.867 − 1.50i)11-s + (−0.403 + 0.698i)12-s + (−0.0793 + 0.0288i)13-s + (−0.0595 − 0.337i)14-s + (−0.293 + 1.66i)15-s + (−0.234 − 0.0855i)16-s + (0.777 − 0.652i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.270632 + 0.708146i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.270632 + 0.708146i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.62 + 0.955i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.407 + 2.30i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.642 + 1.11i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.87 + 4.98i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.285 - 0.104i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.20 + 2.69i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (1.12 - 6.37i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.39 + 2.01i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (3.22 - 5.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + (-4.71 - 1.71i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.171 + 0.974i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (3.36 + 2.82i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.571 + 3.24i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.53 + 2.13i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.90 - 10.7i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.35 + 2.81i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.766 + 4.34i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.13 - 0.775i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (8.05 + 2.93i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.88 + 8.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-14.1 + 5.15i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-13.2 + 11.1i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.33149334444409105146058118222, −9.111602064829413549647610064766, −7.971936012001238261829298538274, −7.13164994910213284676093180915, −5.81753087526124079389022810709, −5.44999772520499958050015472337, −4.66262978573702097743310315124, −3.32122371184360960390644173858, −1.40058078454827014993043454925, −0.41951470628759872093255244573,
2.37419795198683294371296796423, 3.86199473574418882927148244971, 4.82843207375318886508857588477, 5.51887277052974768446938332237, 6.39455230424425167555797124986, 7.15433015802997205907579011714, 7.995334112136121687277726478778, 9.520087258968443567947054596931, 10.48945421066757211860158792357, 10.76818701476180831221890447773
