| L(s) = 1 | + (−0.366 + 1.36i)2-s + (−0.866 − 1.5i)3-s + (−1.73 − i)4-s + (−5.04 − 5.04i)5-s + (2.36 − 0.633i)6-s + (−1.34 − 5.02i)7-s + (2 − 1.99i)8-s + (−1.5 + 2.59i)9-s + (8.74 − 5.04i)10-s + (−7.32 − 1.96i)11-s + 3.46i·12-s + (12.9 − 1.42i)13-s + 7.36·14-s + (−3.19 + 11.9i)15-s + (1.99 + 3.46i)16-s + (−13.9 − 8.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.288 − 0.5i)3-s + (−0.433 − 0.250i)4-s + (−1.00 − 1.00i)5-s + (0.394 − 0.105i)6-s + (−0.192 − 0.718i)7-s + (0.250 − 0.249i)8-s + (−0.166 + 0.288i)9-s + (0.874 − 0.504i)10-s + (−0.665 − 0.178i)11-s + 0.288i·12-s + (0.993 − 0.109i)13-s + 0.525·14-s + (−0.213 + 0.796i)15-s + (0.124 + 0.216i)16-s + (−0.823 − 0.475i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.413291 - 0.462893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.413291 - 0.462893i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 13 | \( 1 + (-12.9 + 1.42i)T \) |
| good | 5 | \( 1 + (5.04 + 5.04i)T + 25iT^{2} \) |
| 7 | \( 1 + (1.34 + 5.02i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (7.32 + 1.96i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (13.9 + 8.08i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-9.87 + 2.64i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (8.29 - 4.78i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-16.5 - 28.7i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (34.2 + 34.2i)T + 961iT^{2} \) |
| 37 | \( 1 + (-63.2 - 16.9i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-14.0 + 52.6i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (40.7 + 23.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-47.8 + 47.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 67.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + (19.4 + 72.7i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (35.2 - 61.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 41.7i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (106. - 28.6i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (36.8 - 36.8i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 13.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-93.1 - 93.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-48.6 - 13.0i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (52.5 - 14.0i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−13.65420465080990901863736648427, −13.07699273258995417750644189957, −11.81135874209172499871172197025, −10.67827587004579113226311145835, −8.991646381230845683881086648887, −8.004506963733916866823379855516, −7.05929287176955074829966941922, −5.48194545019363236179987928244, −4.07782171265526942671399339260, −0.58637852393815146177077860459,
2.90507961410787414679364182459, 4.23739620863164218609186001809, 6.11075728865663418035968654741, 7.75338930906305405507928938430, 8.996232204412030604782297767044, 10.38960668809939143995760449599, 11.15148019309467351798649587782, 11.96224724043970638404432109387, 13.20888708119239437069597812705, 14.67671225652833965630103547360
