L(s) = 1 | + (−0.915 − 0.137i)2-s + (−1.92 + 1.31i)3-s + (−1.09 − 0.337i)4-s + (−0.266 − 3.55i)5-s + (1.94 − 0.934i)6-s + (−0.713 − 2.54i)7-s + (2.62 + 1.26i)8-s + (0.882 − 2.24i)9-s + (−0.246 + 3.29i)10-s + (−1.69 − 4.32i)11-s + (2.54 − 0.784i)12-s + (0.623 − 0.781i)13-s + (0.301 + 2.43i)14-s + (5.17 + 6.49i)15-s + (−0.335 − 0.228i)16-s + (1.38 + 1.28i)17-s + ⋯ |
L(s) = 1 | + (−0.647 − 0.0975i)2-s + (−1.11 + 0.756i)3-s + (−0.546 − 0.168i)4-s + (−0.119 − 1.59i)5-s + (0.792 − 0.381i)6-s + (−0.269 − 0.962i)7-s + (0.926 + 0.446i)8-s + (0.294 − 0.749i)9-s + (−0.0780 + 1.04i)10-s + (−0.511 − 1.30i)11-s + (0.734 − 0.226i)12-s + (0.172 − 0.216i)13-s + (0.0804 + 0.649i)14-s + (1.33 + 1.67i)15-s + (−0.0837 − 0.0571i)16-s + (0.336 + 0.311i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0505424 + 0.206528i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0505424 + 0.206528i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.713 + 2.54i)T \) |
| 13 | \( 1 + (-0.623 + 0.781i)T \) |
good | 2 | \( 1 + (0.915 + 0.137i)T + (1.91 + 0.589i)T^{2} \) |
| 3 | \( 1 + (1.92 - 1.31i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.266 + 3.55i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.69 + 4.32i)T + (-8.06 + 7.48i)T^{2} \) |
| 17 | \( 1 + (-1.38 - 1.28i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.05 + 1.82i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.07 + 0.999i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.284 - 1.24i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.80 + 4.86i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.09 - 1.88i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-0.694 - 0.334i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (5.76 - 2.77i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (0.272 + 0.0410i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-3.68 - 1.13i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.141 + 1.88i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (9.37 - 2.89i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (2.14 - 3.72i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.10 + 4.83i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-13.0 + 1.97i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (3.99 + 6.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.18 - 7.75i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (4.96 - 12.6i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−10.16738709406228071760215354453, −9.296621593460836635606090817090, −8.485934832183748883044633947297, −7.83385266038684566857510188007, −6.17533605003796821848815080749, −5.24318385027074617674237057398, −4.70393405700889723964641655055, −3.76957945316984440354020747706, −1.03573976846576256846019426004, −0.21408147697739868044500363348,
1.92124803205358082379053997438, 3.32655565967555712974385275427, 4.88150698031105731198786494601, 5.90403438944319309905250335792, 6.91410838099274410917298635462, 7.25969853304289783702177544094, 8.338862240768207975206463943856, 9.529977194520867800366456701367, 10.25004368477376224373588661654, 10.94722872334765218549166632417