| L(s) = 1 | + (−8.32 + 3.42i)3-s + (−30.0 − 17.3i)5-s + (−15.6 − 27.0i)7-s + (57.5 − 57.0i)9-s + (−49.9 + 28.8i)11-s + (36.6 − 63.4i)13-s + (309. + 41.4i)15-s + 386. i·17-s − 115.·19-s + (222. + 171. i)21-s + (474. + 274. i)23-s + (290. + 503. i)25-s + (−282. + 671. i)27-s + (−680. + 392. i)29-s + (272. − 471. i)31-s + ⋯ |
| L(s) = 1 | + (−0.924 + 0.380i)3-s + (−1.20 − 0.694i)5-s + (−0.318 − 0.551i)7-s + (0.709 − 0.704i)9-s + (−0.413 + 0.238i)11-s + (0.216 − 0.375i)13-s + (1.37 + 0.184i)15-s + 1.33i·17-s − 0.320·19-s + (0.504 + 0.388i)21-s + (0.897 + 0.518i)23-s + (0.465 + 0.805i)25-s + (−0.388 + 0.921i)27-s + (−0.808 + 0.466i)29-s + (0.283 − 0.490i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.581040 + 0.301010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.581040 + 0.301010i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.32 - 3.42i)T \) |
| good | 5 | \( 1 + (30.0 + 17.3i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (15.6 + 27.0i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (49.9 - 28.8i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-36.6 + 63.4i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 386. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 115.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-474. - 274. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (680. - 392. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-272. + 471. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 898.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.24e3 - 1.29e3i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.00e3 - 1.73e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-702. + 405. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 2.22e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (1.30e3 + 756. i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-951. - 1.64e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.25e3 + 3.90e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 3.99e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.43e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-601. - 1.04e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (8.01e3 - 4.62e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 8.92e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (3.33e3 + 5.77e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−12.66170272578720314187322613323, −11.41657230846332839046406787983, −10.75818784424906247862470369422, −9.610119532480329309439747897684, −8.266241448708943782767076146935, −7.23851503996906681447230631468, −5.86898082948094383433958502185, −4.56749059294113868403036040006, −3.70580722967160254515409121682, −0.887932163109659487177705443573,
0.42294049972860229182528303442, 2.72711895635243261869954406673, 4.34265198817705440088714558250, 5.72587785858315194971460755842, 6.94181461467118142072077370687, 7.66963669802601750990977657147, 9.109795027789512978509517421554, 10.61955851432730501926461399614, 11.32542068034161512331223258034, 12.05853007383559933836963422182
