| L(s) = 1 | + (−0.995 + 0.0950i)2-s + (1.15 − 0.740i)3-s + (0.981 − 0.189i)4-s + (0.312 − 2.17i)5-s + (−1.07 + 0.847i)6-s + (−1.81 − 2.55i)7-s + (−0.959 + 0.281i)8-s + (−0.466 + 1.02i)9-s + (−0.104 + 2.19i)10-s + (3.68 + 2.90i)11-s + (0.991 − 0.945i)12-s + (−1.21 − 5.02i)13-s + (2.05 + 2.37i)14-s + (−1.24 − 2.73i)15-s + (0.928 − 0.371i)16-s + (4.67 + 0.901i)17-s + ⋯ |
| L(s) = 1 | + (−0.703 + 0.0672i)2-s + (0.665 − 0.427i)3-s + (0.490 − 0.0946i)4-s + (0.139 − 0.970i)5-s + (−0.439 + 0.345i)6-s + (−0.687 − 0.965i)7-s + (−0.339 + 0.0996i)8-s + (−0.155 + 0.340i)9-s + (−0.0329 + 0.692i)10-s + (1.11 + 0.874i)11-s + (0.286 − 0.273i)12-s + (−0.338 − 1.39i)13-s + (0.549 + 0.633i)14-s + (−0.322 − 0.705i)15-s + (0.232 − 0.0929i)16-s + (1.13 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.838910 - 0.455471i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.838910 - 0.455471i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.995 - 0.0950i)T \) |
| 67 | \( 1 + (8.17 - 0.447i)T \) |
| good | 3 | \( 1 + (-1.15 + 0.740i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.312 + 2.17i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.81 + 2.55i)T + (-2.28 + 6.61i)T^{2} \) |
| 11 | \( 1 + (-3.68 - 2.90i)T + (2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (1.21 + 5.02i)T + (-11.5 + 5.95i)T^{2} \) |
| 17 | \( 1 + (-4.67 - 0.901i)T + (15.7 + 6.31i)T^{2} \) |
| 19 | \( 1 + (1.52 - 2.14i)T + (-6.21 - 17.9i)T^{2} \) |
| 23 | \( 1 + (-1.77 - 0.915i)T + (13.3 + 18.7i)T^{2} \) |
| 29 | \( 1 + (4.70 - 8.15i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.549 - 2.26i)T + (-27.5 - 14.2i)T^{2} \) |
| 37 | \( 1 + (1.00 + 1.74i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.24 + 9.38i)T + (-32.2 + 25.3i)T^{2} \) |
| 43 | \( 1 + (3.18 - 3.67i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-0.471 - 9.90i)T + (-46.7 + 4.46i)T^{2} \) |
| 53 | \( 1 + (-0.0395 - 0.0456i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.60 - 0.471i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-11.1 + 8.73i)T + (14.3 - 59.2i)T^{2} \) |
| 71 | \( 1 + (-6.31 + 1.21i)T + (65.9 - 26.3i)T^{2} \) |
| 73 | \( 1 + (-3.38 + 2.65i)T + (17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (6.84 - 6.52i)T + (3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-7.11 + 2.84i)T + (60.0 - 57.2i)T^{2} \) |
| 89 | \( 1 + (0.901 + 0.579i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (4.25 + 7.36i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.81248324734461033237819581700, −12.48836814631206187641865498862, −10.72482668817946013319881218863, −9.802277394343024451641477391260, −8.863466135909508846324106211101, −7.81026441433224545500745301481, −6.99140283403260790539623232894, −5.30928751170158291655422796477, −3.44423495741086085421958551330, −1.39371116096327080360088183573,
2.57801132195907435018271618291, 3.66414664971664623195241114916, 6.09197874567330109838917857079, 6.86372092192488474517157423818, 8.517315341062255118976993422079, 9.307680290580630893399256661646, 9.915941105808573547043925140322, 11.43666992445518406902990513841, 11.96667803837413182098308363415, 13.65965341435458116199614978346
