| L(s) = 1 | + (−0.909 + 1.08i)2-s + (1.30 + 2.69i)3-s + (−0.347 − 1.96i)4-s + (−1.54 + 4.23i)5-s + (−4.11 − 1.03i)6-s + (0.223 − 1.26i)7-s + (2.44 + 1.41i)8-s + (−5.56 + 7.06i)9-s + (−3.18 − 5.51i)10-s + (2.33 + 6.41i)11-s + (4.86 − 3.51i)12-s + (8.48 − 7.11i)13-s + (1.17 + 1.39i)14-s + (−13.4 + 1.38i)15-s + (−3.75 + 1.36i)16-s + (24.1 − 13.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.436 + 0.899i)3-s + (−0.0868 − 0.492i)4-s + (−0.308 + 0.846i)5-s + (−0.685 − 0.172i)6-s + (0.0319 − 0.181i)7-s + (0.306 + 0.176i)8-s + (−0.618 + 0.785i)9-s + (−0.318 − 0.551i)10-s + (0.212 + 0.583i)11-s + (0.405 − 0.293i)12-s + (0.652 − 0.547i)13-s + (0.0836 + 0.0997i)14-s + (−0.895 + 0.0923i)15-s + (−0.234 + 0.0855i)16-s + (1.42 − 0.821i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.617149 + 0.794488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.617149 + 0.794488i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 - 1.08i)T \) |
| 3 | \( 1 + (-1.30 - 2.69i)T \) |
| good | 5 | \( 1 + (1.54 - 4.23i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-0.223 + 1.26i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-2.33 - 6.41i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-8.48 + 7.11i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-24.1 + 13.9i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-14.5 + 25.1i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (29.2 - 5.15i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-9.87 + 11.7i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-8.09 - 45.8i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (6.62 + 11.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (27.2 + 32.4i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (48.0 - 17.4i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (57.3 + 10.1i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + 72.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (10.4 - 28.6i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-2.62 + 14.9i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-86.2 + 72.4i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-7.53 + 4.34i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (52.2 - 90.5i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46.1 + 38.7i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (17.6 - 21.0i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-106. - 61.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-40.5 + 14.7i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−15.54828115578211358550599166660, −14.55229844179549121999260263469, −13.74381228706522903235041928447, −11.66749658705114463200491454949, −10.46007648148709435713814311828, −9.619091289731889959669140798799, −8.199907374991123424732009269441, −7.02338648689921961282062273325, −5.18130572162545873090379954646, −3.30698881899406673810695593495,
1.37024545864665462967543842113, 3.63158090751975140815573303641, 6.03008549574743677414520773783, 7.945328665925654414976706011282, 8.534678610424982881194690102291, 9.926760849578660376049497676306, 11.74243690192750804182713851505, 12.28651040396506416132180633494, 13.47195254735235527834170048397, 14.51116359114701765440289605459
