L(s) = 1 | + (−1.71 + 0.246i)3-s + (4.93 + 1.19i)7-s + (2.87 − 0.845i)9-s + (−3.38 − 1.16i)13-s + (−0.539 − 2.22i)19-s + (−8.75 − 0.835i)21-s + (3.27 + 3.77i)25-s + (−4.72 + 2.15i)27-s + (8.77 − 3.03i)31-s + (5.89 + 10.2i)37-s + (6.08 + 1.17i)39-s + (4.69 − 7.31i)43-s + (16.6 + 8.60i)49-s + (1.47 + 3.68i)57-s + (1.00 + 0.712i)61-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.142i)3-s + (1.86 + 0.452i)7-s + (0.959 − 0.281i)9-s + (−0.937 − 0.324i)13-s + (−0.123 − 0.510i)19-s + (−1.91 − 0.182i)21-s + (0.654 + 0.755i)25-s + (−0.909 + 0.415i)27-s + (1.57 − 0.545i)31-s + (0.968 + 1.67i)37-s + (0.974 + 0.187i)39-s + (0.716 − 1.11i)43-s + (2.38 + 1.22i)49-s + (0.195 + 0.487i)57-s + (0.128 + 0.0912i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34666 + 0.220835i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34666 + 0.220835i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.71 - 0.246i)T \) |
| 67 | \( 1 + (-8.18 - 0.186i)T \) |
good | 5 | \( 1 + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-4.93 - 1.19i)T + (6.22 + 3.20i)T^{2} \) |
| 11 | \( 1 + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (3.38 + 1.16i)T + (10.2 + 8.03i)T^{2} \) |
| 17 | \( 1 + (16.9 + 1.61i)T^{2} \) |
| 19 | \( 1 + (0.539 + 2.22i)T + (-16.8 + 8.70i)T^{2} \) |
| 23 | \( 1 + (-5.42 + 22.3i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-8.77 + 3.03i)T + (24.3 - 19.1i)T^{2} \) |
| 37 | \( 1 + (-5.89 - 10.2i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (23.7 - 33.3i)T^{2} \) |
| 43 | \( 1 + (-4.69 + 7.31i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-34.0 - 32.4i)T^{2} \) |
| 53 | \( 1 + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.00 - 0.712i)T + (19.9 + 57.6i)T^{2} \) |
| 71 | \( 1 + (70.6 - 6.74i)T^{2} \) |
| 73 | \( 1 + (6.53 - 9.18i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-0.356 - 1.85i)T + (-73.3 + 29.3i)T^{2} \) |
| 83 | \( 1 + (-81.5 + 15.7i)T^{2} \) |
| 89 | \( 1 + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (1.93 - 1.11i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.47056336168064350586222607187, −9.606125487429205304770719160023, −8.504923816965440192590178367553, −7.72566200797362554676053935283, −6.83824084129934489640138206493, −5.65908144990363488869885984029, −4.95120724621188905979045449559, −4.37002661852931099887251967995, −2.51067961521154217638890476000, −1.14482444600786693838432202829,
1.02981508869601206801733379309, 2.24362751798995147354041188690, 4.31341382714870503495557330541, 4.73617072071392353403210923627, 5.69247185615307465227142054418, 6.78603488612048224043697019921, 7.64591786900580765729860735369, 8.249733510813530159912822027109, 9.534898090131101328704401955315, 10.53499788203272365856295800094