| L(s) = 1 | + (−1.23 − 3.80i)2-s + (7.28 − 5.29i)3-s + (−12.9 + 9.40i)4-s + (−55.7 + 4.47i)5-s + (−29.1 − 21.1i)6-s − 89.1·7-s + (51.7 + 37.6i)8-s + (25.0 − 77.0i)9-s + (85.9 + 206. i)10-s + (−234. − 720. i)11-s + (−44.4 + 136. i)12-s + (−137. + 424. i)13-s + (110. + 339. i)14-s + (−382. + 327. i)15-s + (79.1 − 243. i)16-s + (484. + 352. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.996 + 0.0801i)5-s + (−0.330 − 0.239i)6-s − 0.687·7-s + (0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + (0.271 + 0.652i)10-s + (−0.583 − 1.79i)11-s + (−0.0892 + 0.274i)12-s + (−0.226 + 0.696i)13-s + (0.150 + 0.462i)14-s + (−0.438 + 0.375i)15-s + (0.0772 − 0.237i)16-s + (0.406 + 0.295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8368141611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8368141611\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 + (-7.28 + 5.29i)T \) |
| 5 | \( 1 + (55.7 - 4.47i)T \) |
| good | 7 | \( 1 + 89.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + (234. + 720. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (137. - 424. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-484. - 352. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.87e3 - 1.36e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-1.25e3 - 3.84e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.86e3 - 2.08e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (1.13e3 + 825. i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (4.11e3 - 1.26e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (2.34e3 - 7.21e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 5.60e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.65e4 + 1.20e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.94e4 + 2.13e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (644. - 1.98e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.52e4 - 4.70e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.89e4 + 2.82e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (2.52e4 - 1.83e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.84e3 + 5.68e3i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (2.94e3 - 2.13e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (4.00e4 + 2.91e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-1.01e4 - 3.13e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (1.22e5 - 8.87e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.97073661350219649937372021737, −11.40108413193527370408418261904, −10.18937455275223417912853807389, −9.038320908060944299944803608529, −8.130221482710050638308427610536, −7.19088536824886769476445260064, −5.57368532620343735067822036583, −3.64443743003526436718652888757, −3.09908334792854941460253292922, −1.10264300767076586150567529270,
0.34258065054193079243203485311, 2.76535469663081289601615160290, 4.23967638485663516529100176092, 5.27152008311147831392173438838, 7.17087048509641189025473415732, 7.53704560791712649852775864436, 8.885023756105032129223855303178, 9.780564847376254668583492605944, 10.72440777534780013588803706740, 12.30840483059848075873289544141
