| L(s) = 1 | + (−1.39 + 0.245i)2-s + (1.87 − 0.684i)4-s + (−1.85 + 2.21i)5-s + (2.17 + 0.791i)7-s + (−2.44 + 1.41i)8-s + (2.04 − 3.54i)10-s + (0.401 + 0.478i)11-s + (−4.06 + 23.0i)13-s + (−3.22 − 0.568i)14-s + (3.06 − 2.57i)16-s + (−2.71 − 1.56i)17-s + (2.04 + 3.53i)19-s + (−1.97 + 5.43i)20-s + (−0.675 − 0.567i)22-s + (15.5 + 42.6i)23-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.469 − 0.171i)4-s + (−0.371 + 0.443i)5-s + (0.310 + 0.113i)7-s + (−0.306 + 0.176i)8-s + (0.204 − 0.354i)10-s + (0.0364 + 0.0434i)11-s + (−0.312 + 1.77i)13-s + (−0.230 − 0.0405i)14-s + (0.191 − 0.160i)16-s + (−0.159 − 0.0922i)17-s + (0.107 + 0.186i)19-s + (−0.0989 + 0.271i)20-s + (−0.0307 − 0.0257i)22-s + (0.674 + 1.85i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.563906 + 0.630706i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563906 + 0.630706i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.245i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.85 - 2.21i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-2.17 - 0.791i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (-0.401 - 0.478i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (4.06 - 23.0i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (2.71 + 1.56i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-2.04 - 3.53i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-15.5 - 42.6i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (19.6 - 3.46i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (42.6 - 15.5i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (-18.7 + 32.5i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-45.6 - 8.05i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (-46.3 + 38.9i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (1.06 - 2.91i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 79.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-41.9 + 50.0i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (33.1 + 12.0i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (4.92 - 27.9i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-70.7 - 40.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (12.6 + 21.8i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-11.5 - 65.5i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (49.1 - 8.66i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (111. - 64.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-107. + 90.5i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−12.76542013249984646333719407886, −11.36349063035919682893658268195, −11.22924311330101706907436938407, −9.601129858382553851848700770498, −8.993909198138017345411423182059, −7.53849917953667850276053735999, −6.93313469046160344628526171501, −5.40747601655607369792932425364, −3.74235057906351095423023247334, −1.87540895354809402865519527615,
0.66955704928080735863368177744, 2.76754299153776149183526658389, 4.51100214050722317279578925912, 5.93798360671369686629217366943, 7.46147757389476598644303909523, 8.214002172599149882773956012583, 9.215698213872534147050401662159, 10.44834682568600932009673257294, 11.11660843518016293747564270095, 12.44061773322631183431143587080
