L(s) = 1 | + (−0.149 + 0.360i)2-s + (0.599 + 0.599i)4-s + (0.555 − 0.831i)5-s + (−0.666 + 0.275i)8-s + (0.216 + 0.324i)10-s + 0.566i·16-s + (0.980 − 0.195i)17-s + (0.707 + 0.292i)19-s + (0.831 − 0.165i)20-s + (−1.38 − 0.275i)23-s + (−0.382 − 0.923i)25-s + (−1.08 + 0.216i)31-s + (−0.870 − 0.360i)32-s + (−0.0761 + 0.382i)34-s + (−0.211 + 0.211i)38-s + ⋯ |
L(s) = 1 | + (−0.149 + 0.360i)2-s + (0.599 + 0.599i)4-s + (0.555 − 0.831i)5-s + (−0.666 + 0.275i)8-s + (0.216 + 0.324i)10-s + 0.566i·16-s + (0.980 − 0.195i)17-s + (0.707 + 0.292i)19-s + (0.831 − 0.165i)20-s + (−1.38 − 0.275i)23-s + (−0.382 − 0.923i)25-s + (−1.08 + 0.216i)31-s + (−0.870 − 0.360i)32-s + (−0.0761 + 0.382i)34-s + (−0.211 + 0.211i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.096996746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.096996746\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.555 + 0.831i)T \) |
| 17 | \( 1 + (-0.980 + 0.195i)T \) |
good | 2 | \( 1 + (0.149 - 0.360i)T + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 11 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (1.38 + 0.275i)T + (0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (1.08 - 0.216i)T + (0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (-1.17 - 1.17i)T + iT^{2} \) |
| 53 | \( 1 + (0.750 - 1.81i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (1.81 + 0.750i)T + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (0.382 + 0.923i)T^{2} \) |
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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
−10.50574961551223965595616050325, −9.620119339668463346235752069501, −8.834961216042865038633158842273, −7.951653225828852826004444324565, −7.33563654491416897105606274174, −6.08906083083796749224957334909, −5.54993495817593317331414896236, −4.22652609019042518242815258530, −3.02251001206058180211485285109, −1.70468501177738606497302629244,
1.61598307854116221324870686483, 2.68605633806220370917440214395, 3.71129184390729811805100686376, 5.45327246799181338236401320362, 5.95671534446687698697206915401, 6.98057684572702641245051779590, 7.68768063152714283276267395099, 9.061168475102979998198776710043, 9.949268109691488293317762368774, 10.28781204429219882859497673301