| 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
−14.67671225652833965630103547360, −13.20888708119239437069597812705, −11.96224724043970638404432109387, −11.15148019309467351798649587782, −10.38960668809939143995760449599, −8.996232204412030604782297767044, −7.75338930906305405507928938430, −6.11075728865663418035968654741, −4.23739620863164218609186001809, −2.90507961410787414679364182459,
0.58637852393815146177077860459, 4.07782171265526942671399339260, 5.48194545019363236179987928244, 7.05929287176955074829966941922, 8.004506963733916866823379855516, 8.991646381230845683881086648887, 10.67827587004579113226311145835, 11.81135874209172499871172197025, 13.07699273258995417750644189957, 13.65420465080990901863736648427
