| L(s) = 1 | + (2.11 − 1.43i)2-s + (1.10 − 1.02i)3-s + (1.65 − 4.21i)4-s + (1.18 + 0.366i)5-s + (0.853 − 3.74i)6-s + (−2.52 − 0.798i)7-s + (−1.43 − 6.30i)8-s + (−0.0556 + 0.742i)9-s + (3.03 − 0.937i)10-s + (0.146 + 1.95i)11-s + (−2.48 − 6.33i)12-s + (0.900 + 0.433i)13-s + (−6.47 + 1.94i)14-s + (1.68 − 0.810i)15-s + (−5.46 − 5.07i)16-s + (−2.84 − 0.428i)17-s + ⋯ |
| L(s) = 1 | + (1.49 − 1.01i)2-s + (0.635 − 0.589i)3-s + (0.827 − 2.10i)4-s + (0.531 + 0.164i)5-s + (0.348 − 1.52i)6-s + (−0.953 − 0.301i)7-s + (−0.508 − 2.22i)8-s + (−0.0185 + 0.247i)9-s + (0.961 − 0.296i)10-s + (0.0441 + 0.588i)11-s + (−0.717 − 1.82i)12-s + (0.249 + 0.120i)13-s + (−1.73 + 0.520i)14-s + (0.434 − 0.209i)15-s + (−1.36 − 1.26i)16-s + (−0.690 − 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.06948 - 3.38739i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06948 - 3.38739i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.52 + 0.798i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| good | 2 | \( 1 + (-2.11 + 1.43i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.10 + 1.02i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-1.18 - 0.366i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.146 - 1.95i)T + (-10.8 + 1.63i)T^{2} \) |
| 17 | \( 1 + (2.84 + 0.428i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.682 + 1.18i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.33 + 1.10i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (5.89 - 7.39i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-0.371 - 0.643i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.72 - 6.94i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (1.96 + 8.62i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (0.267 - 1.17i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.330 - 0.225i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-1.43 + 3.66i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (5.14 - 1.58i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (2.46 + 6.27i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-7.34 - 12.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.45 + 5.58i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (6.17 + 4.21i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-1.46 + 2.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (12.2 - 5.90i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.894 + 11.9i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 1.27T + 97T^{2} \) |
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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
−10.56760749756847018636068768363, −9.711619796750593350110392351503, −8.808592580341099073664086821654, −7.18802055211355327198662563179, −6.63149096705818040278745048968, −5.50962464495177522281530186656, −4.54312152469860635676886149461, −3.35002570513191992912967683531, −2.58459940590188485961005792240, −1.58098083710815328364684882690,
2.69441940611950226639508029897, 3.50435065455831068496027762347, 4.30142289773938360286522344841, 5.57996330037506190978001550089, 6.09938398530819873521957176824, 6.98728455313229527702925840256, 8.105206101936726720099705474712, 9.103793185914478774725330544198, 9.653817847798902050811409992592, 11.07049402957378626939351509776
