| L(s) = 1 | + (1.91 − 0.563i)2-s + (−1.96 − 2.26i)3-s + (3.36 − 2.16i)4-s + (−1.52 − 10.5i)5-s + (−5.04 − 3.24i)6-s + (0.567 − 1.24i)7-s + (5.23 − 6.04i)8-s + (−1.28 + 8.90i)9-s + (−8.89 − 19.4i)10-s + (−29.7 − 8.74i)11-s + (−11.5 − 3.38i)12-s + (−8.97 − 19.6i)13-s + (0.388 − 2.70i)14-s + (−21.0 + 24.2i)15-s + (6.64 − 14.5i)16-s + (−87.5 − 56.2i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.136 − 0.947i)5-s + (−0.343 − 0.220i)6-s + (0.0306 − 0.0670i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.281 − 0.615i)10-s + (−0.816 − 0.239i)11-s + (−0.276 − 0.0813i)12-s + (−0.191 − 0.419i)13-s + (0.00741 − 0.0516i)14-s + (−0.362 + 0.417i)15-s + (0.103 − 0.227i)16-s + (−1.24 − 0.803i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.670906 - 1.55001i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.670906 - 1.55001i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.91 + 0.563i)T \) |
| 3 | \( 1 + (1.96 + 2.26i)T \) |
| 23 | \( 1 + (-101. + 44.1i)T \) |
| good | 5 | \( 1 + (1.52 + 10.5i)T + (-119. + 35.2i)T^{2} \) |
| 7 | \( 1 + (-0.567 + 1.24i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (29.7 + 8.74i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (8.97 + 19.6i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (87.5 + 56.2i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-11.8 + 7.58i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (-24.5 - 15.7i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (-44.7 + 51.5i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (0.861 - 5.99i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-14.3 - 99.7i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-186. - 214. i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 - 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-147. + 323. i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (187. + 410. i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (25.9 - 29.9i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (-279. + 82.0i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-782. + 229. i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (348. - 223. i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (-471. - 1.03e3i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-150. + 1.04e3i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (275. + 318. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-110. - 768. i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
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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
−12.60378839776398618733837112810, −11.48052172759457317012420308781, −10.63990975646682640699819035324, −9.172815571028455976022380310537, −7.959903952566993981577072108735, −6.73609308709771299972893715743, −5.36561323416706622332807943314, −4.56642014298317892223474889928, −2.62418817661246928588681461619, −0.70106552626400861668856822791,
2.55576480051568054990859409057, 3.99576936662575653939707831454, 5.24428738140380890884964845702, 6.50098039634037949999424319394, 7.40218261489806472020789477879, 8.919679235533031059113383510702, 10.44369753145621991321142040664, 10.97790319496976085365832213052, 12.07004740166799650517939267686, 13.14676597475746398218664929012
