| L(s) = 1 | + (0.967 + 0.379i)2-s + (−1.05 + 5.08i)3-s + (−5.07 − 4.70i)4-s + (4.76 + 3.25i)5-s + (−2.95 + 4.52i)6-s + (−12.8 − 13.3i)7-s + (−6.73 − 13.9i)8-s + (−24.7 − 10.7i)9-s + (3.38 + 4.95i)10-s + (−2.97 − 19.7i)11-s + (29.3 − 20.8i)12-s + (46.4 − 37.0i)13-s + (−7.40 − 17.7i)14-s + (−21.5 + 20.8i)15-s + (2.93 + 39.1i)16-s + (−12.5 − 3.87i)17-s + ⋯ |
| L(s) = 1 | + (0.342 + 0.134i)2-s + (−0.203 + 0.979i)3-s + (−0.633 − 0.588i)4-s + (0.426 + 0.290i)5-s + (−0.201 + 0.307i)6-s + (−0.695 − 0.718i)7-s + (−0.297 − 0.617i)8-s + (−0.917 − 0.398i)9-s + (0.106 + 0.156i)10-s + (−0.0815 − 0.541i)11-s + (0.705 − 0.500i)12-s + (0.991 − 0.790i)13-s + (−0.141 − 0.339i)14-s + (−0.371 + 0.358i)15-s + (0.0457 + 0.611i)16-s + (−0.178 − 0.0552i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0390 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0390 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.658319 - 0.633075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.658319 - 0.633075i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.05 - 5.08i)T \) |
| 7 | \( 1 + (12.8 + 13.3i)T \) |
| good | 2 | \( 1 + (-0.967 - 0.379i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-4.76 - 3.25i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (2.97 + 19.7i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-46.4 + 37.0i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (12.5 + 3.87i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (8.14 - 4.70i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (39.2 + 127. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-14.9 + 3.40i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (149. + 86.5i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (234. - 217. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-293. + 141. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (140. + 67.7i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (28.1 - 71.6i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (36.8 - 39.6i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (721. - 491. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (235. + 253. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-161. + 279. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-906. - 206. i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (195. - 76.5i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (62.0 + 107. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (170. - 213. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-817. - 123. i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 1.01e3iT - 9.12e5T^{2} \) |
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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
−12.55654165136483186580718617782, −10.83748294003938341844352947082, −10.39721927507967679602652210277, −9.474442973473471836411158957897, −8.423044015484913445550115299789, −6.41304655713970075119383053403, −5.73831425934262631414191848774, −4.36863125711336274164698575239, −3.32072172155124713927835625174, −0.40778561299270935708012975033,
1.90942321674313177976640751504, 3.50192937289347690022474176670, 5.21968879361560828407117398638, 6.22131384303658764024290875699, 7.51147722410440437113040917976, 8.770578044092914565009925931370, 9.405470572399397949363755798774, 11.21861288972167447611117988291, 12.15449557901254726316765528056, 12.88300783989171622767709834104
