| L(s) = 1 | + (−1.37 + 1.64i)3-s + (3.12 − 1.13i)5-s + (5.96 + 10.3i)7-s + (0.762 + 4.32i)9-s + (−2.15 + 3.73i)11-s + (−4.95 − 5.90i)13-s + (−2.44 + 6.71i)15-s + (4.82 − 27.3i)17-s + (13.2 − 13.6i)19-s + (−25.2 − 4.44i)21-s + (3.33 + 1.21i)23-s + (−10.6 + 8.95i)25-s + (−24.8 − 14.3i)27-s + (14.6 − 2.57i)29-s + (−25.3 + 14.6i)31-s + ⋯ |
| L(s) = 1 | + (−0.459 + 0.548i)3-s + (0.625 − 0.227i)5-s + (0.852 + 1.47i)7-s + (0.0847 + 0.480i)9-s + (−0.195 + 0.339i)11-s + (−0.381 − 0.454i)13-s + (−0.162 + 0.447i)15-s + (0.283 − 1.60i)17-s + (0.697 − 0.716i)19-s + (−1.20 − 0.211i)21-s + (0.144 + 0.0527i)23-s + (−0.426 + 0.358i)25-s + (−0.922 − 0.532i)27-s + (0.503 − 0.0888i)29-s + (−0.819 + 0.473i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.10458 + 0.568535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10458 + 0.568535i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-13.2 + 13.6i)T \) |
| good | 3 | \( 1 + (1.37 - 1.64i)T + (-1.56 - 8.86i)T^{2} \) |
| 5 | \( 1 + (-3.12 + 1.13i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-5.96 - 10.3i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (2.15 - 3.73i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (4.95 + 5.90i)T + (-29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-4.82 + 27.3i)T + (-271. - 98.8i)T^{2} \) |
| 23 | \( 1 + (-3.33 - 1.21i)T + (405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-14.6 + 2.57i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (25.3 - 14.6i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 32.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-11.1 + 13.3i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-1.29 + 0.472i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-0.165 - 0.936i)T + (-2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + (-26.6 + 73.2i)T + (-2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-95.8 - 16.8i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-36.0 - 13.1i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (84.5 - 14.9i)T + (4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (48.2 + 132. i)T + (-3.86e3 + 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-81.3 - 68.2i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (15.2 - 18.1i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (64.1 + 111. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-92.0 - 109. i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (135. + 23.8i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−14.53456065956569873984752349233, −13.35325178646587637243843749510, −12.04721110072895992291443525912, −11.23316689197491329915996782008, −9.884882732458802659919052095681, −8.968223995641001940886105237648, −7.50065042948205606677937574655, −5.41876928628903351564698355176, −5.08695863513987362042129259261, −2.37837686155140660564275706781,
1.38805306434807013953050731852, 4.00797155567794187701582247192, 5.79611601513785363833576828992, 6.98115780117862131367136886128, 8.087284806067961017577094214772, 9.870729173668227530126972234718, 10.77496362868734632735312050035, 11.89918417236379824655896898215, 13.10178455182749607414677786587, 14.04953264713197473706504720320
