| L(s) = 1 | + (1.45 + 1.37i)2-s + (1.35 + 1.07i)3-s + (0.232 + 3.99i)4-s + (4.27 + 2.05i)5-s + (0.487 + 3.42i)6-s + (4.43 + 3.53i)7-s + (−5.14 + 6.12i)8-s + (0.667 + 2.92i)9-s + (3.39 + 8.86i)10-s + (−12.7 − 2.90i)11-s + (−3.99 + 5.65i)12-s + (1.53 − 6.73i)13-s + (1.59 + 11.2i)14-s + (3.56 + 7.40i)15-s + (−15.8 + 1.85i)16-s + 14.0·17-s + ⋯ |
| L(s) = 1 | + (0.727 + 0.686i)2-s + (0.451 + 0.359i)3-s + (0.0582 + 0.998i)4-s + (0.855 + 0.411i)5-s + (0.0813 + 0.571i)6-s + (0.633 + 0.505i)7-s + (−0.642 + 0.766i)8-s + (0.0741 + 0.324i)9-s + (0.339 + 0.886i)10-s + (−1.15 − 0.264i)11-s + (−0.333 + 0.471i)12-s + (0.118 − 0.517i)13-s + (0.114 + 0.802i)14-s + (0.237 + 0.493i)15-s + (−0.993 + 0.116i)16-s + 0.824·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.421 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.73498 + 2.71819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73498 + 2.71819i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.45 - 1.37i)T \) |
| 3 | \( 1 + (-1.35 - 1.07i)T \) |
| 29 | \( 1 + (-25.3 + 14.0i)T \) |
| good | 5 | \( 1 + (-4.27 - 2.05i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (-4.43 - 3.53i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (12.7 + 2.90i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (-1.53 + 6.73i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 - 14.0T + 289T^{2} \) |
| 19 | \( 1 + (-12.0 + 9.57i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (4.81 + 10.0i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (9.45 - 19.6i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (-8.78 - 38.4i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 + 65.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-4.21 - 8.75i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (-15.6 - 3.57i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (-6.96 - 3.35i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 26.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (-12.3 + 15.4i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (-91.8 + 20.9i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-59.2 - 13.5i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-24.3 + 11.7i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (-104. + 23.7i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (-18.8 + 15.0i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (126. + 60.9i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (-13.8 - 17.3i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.70299075967322475096596799544, −10.59107728716485023073598251594, −9.733222222061951816841353600462, −8.415865209488652508847324103214, −7.933645128441566064386537180899, −6.62622800453238054113959105803, −5.49919476406181262401046065517, −4.94058042250917834057235476106, −3.27463531633846930979028411216, −2.38139157893076253752366886015,
1.26404544614452269702227882379, 2.29486802690805064025207509520, 3.69245534607684887200892475153, 5.02167356748830640683947445893, 5.73693456837453385798923753793, 7.11952283573298684678628802164, 8.164875464680279930152136175506, 9.452993741051128902661871669355, 10.09100784296367361760668539728, 11.03257743463908597229129932322
