| L(s) = 1 | + (−2.50 + 0.378i)2-s + (2.68 + 1.83i)3-s + (4.24 − 1.30i)4-s + (−0.0815 + 1.08i)5-s + (−7.43 − 3.57i)6-s + (−1.85 + 1.88i)7-s + (−5.57 + 2.68i)8-s + (2.76 + 7.04i)9-s + (−0.206 − 2.75i)10-s + (1.58 − 4.03i)11-s + (13.7 + 4.25i)12-s + (0.623 + 0.781i)13-s + (3.94 − 5.43i)14-s + (−2.21 + 2.77i)15-s + (5.63 − 3.84i)16-s + (−2.71 + 2.51i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 + 0.267i)2-s + (1.55 + 1.05i)3-s + (2.12 − 0.654i)4-s + (−0.0364 + 0.486i)5-s + (−3.03 − 1.46i)6-s + (−0.701 + 0.712i)7-s + (−1.97 + 0.948i)8-s + (0.921 + 2.34i)9-s + (−0.0653 − 0.872i)10-s + (0.477 − 1.21i)11-s + (3.97 + 1.22i)12-s + (0.172 + 0.216i)13-s + (1.05 − 1.45i)14-s + (−0.570 + 0.715i)15-s + (1.40 − 0.960i)16-s + (−0.657 + 0.610i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.245454 + 0.904400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.245454 + 0.904400i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.85 - 1.88i)T \) |
| 13 | \( 1 + (-0.623 - 0.781i)T \) |
| good | 2 | \( 1 + (2.50 - 0.378i)T + (1.91 - 0.589i)T^{2} \) |
| 3 | \( 1 + (-2.68 - 1.83i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (0.0815 - 1.08i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.58 + 4.03i)T + (-8.06 - 7.48i)T^{2} \) |
| 17 | \( 1 + (2.71 - 2.51i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (3.05 - 5.29i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.25 + 3.02i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.314 + 1.37i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.21 - 2.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.25 + 1.00i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (6.85 - 3.30i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-7.77 - 3.74i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-10.8 + 1.63i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-1.93 + 0.596i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.0449 + 0.599i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (6.71 + 2.07i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-2.94 - 5.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.82 + 12.3i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-12.5 - 1.88i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (0.0838 - 0.145i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.64 - 12.0i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (2.38 + 6.07i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 2.99T + 97T^{2} \) |
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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
−10.52355590267155785646474066566, −9.827796577865524957638513037947, −9.022485173791062452545854292528, −8.560204969495880387728347619323, −8.070761190882778188834491596777, −6.78775232977036759086630228983, −5.94626037745452727775405621588, −3.97303044688149166764895948574, −2.97501019537079923750334590095, −2.01227473975092038989577802275,
0.73486292311173030837222796219, 1.92091883539759236996176281525, 2.79842312332353348164330211113, 4.11873194956903063261922146465, 6.66153523781064646094263176064, 7.06414835989474218085307615215, 7.70495635614363317440154229017, 8.728133211332888904955480527686, 9.126957458118791107348966059483, 9.767725132840976600608308193278
