| 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
−14.04953264713197473706504720320, −13.10178455182749607414677786587, −11.89918417236379824655896898215, −10.77496362868734632735312050035, −9.870729173668227530126972234718, −8.087284806067961017577094214772, −6.98115780117862131367136886128, −5.79611601513785363833576828992, −4.00797155567794187701582247192, −1.38805306434807013953050731852,
2.37837686155140660564275706781, 5.08695863513987362042129259261, 5.41876928628903351564698355176, 7.50065042948205606677937574655, 8.968223995641001940886105237648, 9.884882732458802659919052095681, 11.23316689197491329915996782008, 12.04721110072895992291443525912, 13.35325178646587637243843749510, 14.53456065956569873984752349233
