L(s) = 1 | + (0.928 + 0.371i)3-s + (−0.888 − 0.458i)4-s + (−0.959 − 0.281i)7-s + (0.723 + 0.690i)9-s + (−0.654 − 0.755i)12-s + (0.627 − 1.81i)13-s + (0.580 + 0.814i)16-s + (1.34 − 1.28i)19-s + (−0.786 − 0.618i)21-s + (0.981 − 0.189i)25-s + (0.415 + 0.909i)27-s + (0.723 + 0.690i)28-s + (0.815 + 0.157i)31-s + (−0.327 − 0.945i)36-s + (−0.981 + 1.70i)37-s + ⋯ |
L(s) = 1 | + (0.928 + 0.371i)3-s + (−0.888 − 0.458i)4-s + (−0.959 − 0.281i)7-s + (0.723 + 0.690i)9-s + (−0.654 − 0.755i)12-s + (0.627 − 1.81i)13-s + (0.580 + 0.814i)16-s + (1.34 − 1.28i)19-s + (−0.786 − 0.618i)21-s + (0.981 − 0.189i)25-s + (0.415 + 0.909i)27-s + (0.723 + 0.690i)28-s + (0.815 + 0.157i)31-s + (−0.327 − 0.945i)36-s + (−0.981 + 1.70i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.163629034\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.163629034\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.928 - 0.371i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 67 | \( 1 + (0.654 + 0.755i)T \) |
good | 2 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 5 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 11 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 13 | \( 1 + (-0.627 + 1.81i)T + (-0.786 - 0.618i)T^{2} \) |
| 17 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 19 | \( 1 + (-1.34 + 1.28i)T + (0.0475 - 0.998i)T^{2} \) |
| 23 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.815 - 0.157i)T + (0.928 + 0.371i)T^{2} \) |
| 37 | \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 43 | \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 53 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 59 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 61 | \( 1 + (1.84 + 0.176i)T + (0.981 + 0.189i)T^{2} \) |
| 71 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 73 | \( 1 + (-0.481 - 1.05i)T + (-0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (1.28 + 1.48i)T + (-0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 89 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 97 | \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.691572705464969766256542668096, −8.895162344668201745095811083946, −8.333775459610739160048861554084, −7.41381827653819450295508077362, −6.41108848118558730858583095346, −5.28568006078724410711279675922, −4.65780341014906032101540080814, −3.32396991597532803579940631019, −3.06475830124788027631720265435, −1.03422754971839484393639144782,
1.53762698851530422074182184814, 2.99613063082957119472777242131, 3.66626703284685967523122038422, 4.46727318736973282330911809253, 5.78186853194452554596997669804, 6.75625854235887799410748934938, 7.45385361985801022030540917961, 8.415666843045264635723887632821, 9.044399953516351832426517179894, 9.490918917448901542317092363451