| L(s) = 1 | + (−1.23 − 3.80i)2-s + (7.28 − 5.29i)3-s + (−12.9 + 9.40i)4-s + (−49.2 − 26.3i)5-s + (−29.1 − 21.1i)6-s + 255.·7-s + (51.7 + 37.6i)8-s + (25.0 − 77.0i)9-s + (−39.4 + 220. i)10-s + (149. + 458. i)11-s + (−44.4 + 136. i)12-s + (−307. + 947. i)13-s + (−315. − 970. i)14-s + (−498. + 68.6i)15-s + (79.1 − 243. i)16-s + (87.9 + 63.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.881 − 0.471i)5-s + (−0.330 − 0.239i)6-s + 1.96·7-s + (0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.124 + 0.696i)10-s + (0.371 + 1.14i)11-s + (−0.0892 + 0.274i)12-s + (−0.505 + 1.55i)13-s + (−0.429 − 1.32i)14-s + (−0.571 + 0.0787i)15-s + (0.0772 − 0.237i)16-s + (0.0737 + 0.0535i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.084035993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084035993\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (49.2 + 26.3i)T \) |
| good | 7 | \( 1 - 255.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-149. - 458. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (307. - 947. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-87.9 - 63.8i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-673. - 489. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (343. + 1.05e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-5.86e3 + 4.26e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.32e3 - 2.41e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.35e3 + 7.23e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.15e3 - 3.54e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 5.70e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.99e4 - 1.44e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.92e4 + 1.39e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-5.40e3 + 1.66e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (6.69e3 + 2.06e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-4.08e4 - 2.96e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (1.44e4 - 1.04e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.96e3 - 6.04e3i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.86e4 - 5.71e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (9.73e4 + 7.07e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.82e4 + 5.60e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (8.09e4 - 5.88e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.77078803954847113050759785582, −11.46377395064570857223913840219, −9.910535940278797942499070902508, −8.750793262381104571048693558434, −8.010550755363953899452118989381, −7.11783270365359785850973404180, −4.72774478670009246952555391395, −4.23976773184880341749998344624, −2.16634789832822187551223956420, −1.21207556310711290220429011123,
0.907338888249547177497408170358, 3.03886371024491475824617382482, 4.49640247837479836993189816219, 5.50168551323110553082153496806, 7.25544681677797805160929278412, 8.169193705401386107410198368337, 8.517014040377094007837896078476, 10.24495104507827068278662942667, 11.09394105696193819096879521327, 11.96202312918236600340643601024
