| L(s) = 1 | + (1.46 − 1.00i)2-s + (−0.670 + 0.622i)3-s + (0.421 − 1.07i)4-s + (−3.65 − 1.12i)5-s + (−0.361 + 1.58i)6-s + (2.44 + 1.02i)7-s + (0.333 + 1.46i)8-s + (−0.161 + 2.15i)9-s + (−6.49 + 2.00i)10-s + (0.0318 + 0.425i)11-s + (0.386 + 0.983i)12-s + (−0.900 − 0.433i)13-s + (4.60 − 0.944i)14-s + (3.15 − 1.51i)15-s + (3.64 + 3.38i)16-s + (3.25 + 0.490i)17-s + ⋯ |
| L(s) = 1 | + (1.03 − 0.707i)2-s + (−0.387 + 0.359i)3-s + (0.210 − 0.537i)4-s + (−1.63 − 0.504i)5-s + (−0.147 + 0.646i)6-s + (0.922 + 0.385i)7-s + (0.118 + 0.517i)8-s + (−0.0538 + 0.719i)9-s + (−2.05 + 0.634i)10-s + (0.00961 + 0.128i)11-s + (0.111 + 0.283i)12-s + (−0.249 − 0.120i)13-s + (1.23 − 0.252i)14-s + (0.814 − 0.392i)15-s + (0.911 + 0.846i)16-s + (0.789 + 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38799 + 0.685716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38799 + 0.685716i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.44 - 1.02i)T \) |
| 13 | \( 1 + (0.900 + 0.433i)T \) |
| good | 2 | \( 1 + (-1.46 + 1.00i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (0.670 - 0.622i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (3.65 + 1.12i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.0318 - 0.425i)T + (-10.8 + 1.63i)T^{2} \) |
| 17 | \( 1 + (-3.25 - 0.490i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (3.69 - 6.40i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.14 + 0.323i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (5.11 - 6.41i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-5.20 - 9.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.40 + 6.13i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (2.34 + 10.2i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.221 + 0.968i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.101 - 0.0694i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-2.41 + 6.14i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (7.16 - 2.21i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (3.28 + 8.37i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-5.47 - 9.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.132 - 0.166i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (3.73 + 2.54i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (0.506 - 0.877i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.97 + 4.80i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.587 - 7.84i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−10.87996432441590399649555985764, −10.49800891023659578543419279533, −8.684100506460417629838800713468, −8.148096602419794726196532773006, −7.37588916436666273379204805453, −5.49852540413159142539750537707, −4.99807080166049429967184439147, −4.17244105421256733518309202411, −3.39111190064194628216343425562, −1.80004244392790057729086454526,
0.66524491694160411959411354906, 3.12462810891123125261802264860, 4.19855075203138699985334670864, 4.71266293253569086901572039521, 6.02513565536363343236607352737, 6.86429922917854262278392082251, 7.52002476662058402372184271681, 8.196388268775919743386503133681, 9.592713993335507233548111032591, 10.87342230673373117548880296597
