L(s) = 1 | + 3-s + 5-s + (−0.436 − 0.756i)7-s + 9-s + (1.09 + 1.89i)11-s + (−1.27 + 2.20i)13-s + 15-s + (−3.07 + 5.33i)17-s + (−1.66 + 2.89i)19-s + (−0.436 − 0.756i)21-s + (−2.10 + 3.64i)23-s + 25-s + 27-s + (−0.323 − 0.559i)29-s + (0.579 + 1.00i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + (−0.165 − 0.285i)7-s + 0.333·9-s + (0.330 + 0.572i)11-s + (−0.353 + 0.611i)13-s + 0.258·15-s + (−0.746 + 1.29i)17-s + (−0.382 + 0.663i)19-s + (−0.0953 − 0.165i)21-s + (−0.438 + 0.759i)23-s + 0.200·25-s + 0.192·27-s + (−0.0600 − 0.103i)29-s + (0.104 + 0.180i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.433557946\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.433557946\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-1.38 - 8.06i)T \) |
good | 7 | \( 1 + (0.436 + 0.756i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.09 - 1.89i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.27 - 2.20i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.07 - 5.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.66 - 2.89i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.10 - 3.64i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.323 + 0.559i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.579 - 1.00i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.00 + 6.93i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 9.15T + 43T^{2} \) |
| 47 | \( 1 + (6.21 + 10.7i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.66T + 53T^{2} \) |
| 59 | \( 1 + 3.64T + 59T^{2} \) |
| 61 | \( 1 + (2.99 - 5.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-5.31 - 9.20i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.0239 - 0.0414i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.57 + 2.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.29 - 3.97i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.07T + 89T^{2} \) |
| 97 | \( 1 + (-1.76 + 3.06i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.601339211930500387364972088890, −8.166246572575848603043488857438, −7.05910843354694195929504881107, −6.70189644700841956782056634604, −5.81107404189459967196921735580, −4.83905166782850015322815753269, −4.01732178892513985278401391628, −3.39881346689295320251949071929, −2.04240033240336090048071692791, −1.66674413900721078566571543655,
0.33692333150632903853650899980, 1.77066685599394709168343359423, 2.74726727943431886978661498159, 3.24963581614992514927976498737, 4.54404362350840407589444518373, 5.03033291700224657570173334032, 6.18609518299832888026411605458, 6.60472569713894426472938789987, 7.56131476896299887865495764987, 8.251975496014425298418104033476