L(s) = 1 | + (0.212 − 0.977i)2-s + (−0.800 + 0.599i)3-s + (−0.909 − 0.415i)4-s + (0.475 + 2.18i)5-s + (0.415 + 0.909i)6-s + (−0.0253 + 0.354i)7-s + (−0.599 + 0.800i)8-s + (0.281 − 0.959i)9-s + (2.23 − 0.000365i)10-s + (0.177 − 0.276i)11-s + (0.977 − 0.212i)12-s + (−3.66 + 0.262i)13-s + (0.341 + 0.100i)14-s + (−1.69 − 1.46i)15-s + (0.654 + 0.755i)16-s + (−0.571 + 1.53i)17-s + ⋯ |
L(s) = 1 | + (0.150 − 0.690i)2-s + (−0.462 + 0.345i)3-s + (−0.454 − 0.207i)4-s + (0.212 + 0.977i)5-s + (0.169 + 0.371i)6-s + (−0.00958 + 0.134i)7-s + (−0.211 + 0.283i)8-s + (0.0939 − 0.319i)9-s + (0.707 − 0.000115i)10-s + (0.0536 − 0.0834i)11-s + (0.282 − 0.0613i)12-s + (−1.01 + 0.0727i)13-s + (0.0911 + 0.0267i)14-s + (−0.436 − 0.378i)15-s + (0.163 + 0.188i)16-s + (−0.138 + 0.371i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.463197 + 0.583522i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.463197 + 0.583522i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.212 + 0.977i)T \) |
| 3 | \( 1 + (0.800 - 0.599i)T \) |
| 5 | \( 1 + (-0.475 - 2.18i)T \) |
| 23 | \( 1 + (-0.328 + 4.78i)T \) |
good | 7 | \( 1 + (0.0253 - 0.354i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (-0.177 + 0.276i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.66 - 0.262i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (0.571 - 1.53i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (2.28 - 5.01i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (7.40 - 3.38i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.10 - 7.66i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.95 + 5.41i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (3.28 - 0.965i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-5.70 - 7.62i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (7.08 - 7.08i)T - 47iT^{2} \) |
| 53 | \( 1 + (10.7 + 0.770i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-4.22 - 3.66i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-10.9 + 1.57i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (0.320 + 0.0697i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-13.0 + 8.37i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.56 + 2.44i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (2.77 - 3.19i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (12.4 - 6.80i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-0.331 + 2.30i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (8.52 + 4.65i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.78142146335996159239432352105, −10.06084041476653686352179766212, −9.372839033411780175693884078083, −8.214298333104699858202950048508, −7.05100462690102675119407678586, −6.16318322007964772008528127770, −5.22390849165779514869537852670, −4.11097526544373482664993378610, −3.09177961398194587187187666437, −1.91003287901095560243991331978,
0.37882183183238675125639045247, 2.16354325186117780754782814868, 4.00710544241584048943149621548, 5.01813155093635131512227244151, 5.57232697436048558211122958063, 6.73719095405654257945988842307, 7.47422163030443203819733128927, 8.359543297178848921709414761527, 9.348078614921400837159676043840, 9.923935457780560584309486935028