| L(s) = 1 | + (0.376 − 1.36i)2-s + (−1.71 − 1.02i)4-s + (−1.29 + 3.56i)5-s + (−2.58 − 0.455i)7-s + (−2.04 + 1.95i)8-s + (4.37 + 3.10i)10-s + (−3.39 + 1.23i)11-s + (−0.819 + 0.688i)13-s + (−1.59 + 3.34i)14-s + (1.89 + 3.52i)16-s + (−0.980 + 0.566i)17-s + (0.0627 + 0.0362i)19-s + (5.88 − 4.78i)20-s + (0.407 + 5.09i)22-s + (0.0731 + 0.414i)23-s + ⋯ |
| L(s) = 1 | + (0.265 − 0.963i)2-s + (−0.858 − 0.512i)4-s + (−0.580 + 1.59i)5-s + (−0.975 − 0.171i)7-s + (−0.722 + 0.691i)8-s + (1.38 + 0.983i)10-s + (−1.02 + 0.372i)11-s + (−0.227 + 0.190i)13-s + (−0.425 + 0.894i)14-s + (0.474 + 0.880i)16-s + (−0.237 + 0.137i)17-s + (0.0143 + 0.00830i)19-s + (1.31 − 1.07i)20-s + (0.0868 + 1.08i)22-s + (0.0152 + 0.0864i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0709 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0709 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.309317 + 0.332103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.309317 + 0.332103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.376 + 1.36i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.29 - 3.56i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.58 + 0.455i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (3.39 - 1.23i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.819 - 0.688i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.980 - 0.566i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0627 - 0.0362i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0731 - 0.414i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.192 + 0.229i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.88 + 1.21i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.377 - 0.654i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.36 + 7.58i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.18 - 8.74i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0443 - 0.251i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 12.0iT - 53T^{2} \) |
| 59 | \( 1 + (11.7 + 4.27i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.13 - 6.43i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.37 + 5.21i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.35 - 11.0i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.578 - 1.00i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.09 - 6.07i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.00 - 2.51i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.07 + 1.19i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.81 + 0.660i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.76820568284208182530082055777, −10.82146303775294085149683704950, −10.29365785170311791998310057543, −9.510929518002002234085049208484, −8.043639346625871731903539815104, −6.98507111045710468341941642972, −6.00221569815985630429978630043, −4.44617048388389940625010479920, −3.26725741700611758907547915100, −2.54370605002370770730240728355,
0.28495619223546754564295045458, 3.24414948565022818216903996194, 4.57557855674543121297606825826, 5.31564524097326884941169476280, 6.40544345883154104759566993825, 7.69229253279226372786202029585, 8.413507902313199011000187582933, 9.175193652582607367090828422376, 10.13297648479152077792179612203, 11.78740682776233569205064237534
