| L(s) = 1 | + (−2.82 − 12.8i)2-s + (−6.70 + 40.9i)3-s + (−98.2 + 45.4i)4-s + (−36.2 + 130. i)5-s + (543. − 29.4i)6-s + (400. − 379. i)7-s + (351. + 462. i)8-s + (−938. − 316. i)9-s + (1.77e3 + 96.3i)10-s + (−1.42e3 + 1.20e3i)11-s + (−1.20e3 − 4.32e3i)12-s + (15.4 + 45.9i)13-s + (−5.99e3 − 4.06e3i)14-s + (−5.10e3 − 2.35e3i)15-s + (454. − 534. i)16-s + (−404. − 382. i)17-s + ⋯ |
| L(s) = 1 | + (−0.352 − 1.60i)2-s + (−0.248 + 1.51i)3-s + (−1.53 + 0.710i)4-s + (−0.290 + 1.04i)5-s + (2.51 − 0.136i)6-s + (1.16 − 1.10i)7-s + (0.687 + 0.904i)8-s + (−1.28 − 0.433i)9-s + (1.77 + 0.0963i)10-s + (−1.06 + 0.906i)11-s + (−0.695 − 2.50i)12-s + (0.00704 + 0.0209i)13-s + (−2.18 − 1.48i)14-s + (−1.51 − 0.699i)15-s + (0.110 − 0.130i)16-s + (−0.0822 − 0.0779i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0122277 + 0.123044i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0122277 + 0.123044i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (1.28e5 - 1.60e5i)T \) |
| good | 2 | \( 1 + (2.82 + 12.8i)T + (-58.0 + 26.8i)T^{2} \) |
| 3 | \( 1 + (6.70 - 40.9i)T + (-690. - 232. i)T^{2} \) |
| 5 | \( 1 + (36.2 - 130. i)T + (-1.33e4 - 8.05e3i)T^{2} \) |
| 7 | \( 1 + (-400. + 379. i)T + (6.36e3 - 1.17e5i)T^{2} \) |
| 11 | \( 1 + (1.42e3 - 1.20e3i)T + (2.86e5 - 1.74e6i)T^{2} \) |
| 13 | \( 1 + (-15.4 - 45.9i)T + (-3.84e6 + 2.92e6i)T^{2} \) |
| 17 | \( 1 + (404. + 382. i)T + (1.30e6 + 2.41e7i)T^{2} \) |
| 19 | \( 1 + (3.05e3 + 7.67e3i)T + (-3.41e7 + 3.23e7i)T^{2} \) |
| 23 | \( 1 + (2.12e3 + 1.95e4i)T + (-1.44e8 + 3.18e7i)T^{2} \) |
| 29 | \( 1 + (2.56e4 + 5.63e3i)T + (5.39e8 + 2.49e8i)T^{2} \) |
| 31 | \( 1 + (5.17e4 + 2.06e4i)T + (6.44e8 + 6.10e8i)T^{2} \) |
| 37 | \( 1 + (-3.08e4 + 4.05e4i)T + (-6.86e8 - 2.47e9i)T^{2} \) |
| 41 | \( 1 + (5.96e4 + 6.48e3i)T + (4.63e9 + 1.02e9i)T^{2} \) |
| 43 | \( 1 + (-5.61e4 - 4.77e4i)T + (1.02e9 + 6.23e9i)T^{2} \) |
| 47 | \( 1 + (-5.18e4 + 1.43e4i)T + (9.23e9 - 5.55e9i)T^{2} \) |
| 53 | \( 1 + (-3.03e3 - 5.60e4i)T + (-2.20e10 + 2.39e9i)T^{2} \) |
| 61 | \( 1 + (-5.19e3 - 2.35e4i)T + (-4.67e10 + 2.16e10i)T^{2} \) |
| 67 | \( 1 + (1.01e5 + 1.33e5i)T + (-2.42e10 + 8.71e10i)T^{2} \) |
| 71 | \( 1 + (-1.89e4 - 6.82e4i)T + (-1.09e11 + 6.60e10i)T^{2} \) |
| 73 | \( 1 + (4.13e5 + 2.80e5i)T + (5.60e10 + 1.40e11i)T^{2} \) |
| 79 | \( 1 + (-2.67e4 - 1.63e5i)T + (-2.30e11 + 7.76e10i)T^{2} \) |
| 83 | \( 1 + (-1.38e5 + 7.34e4i)T + (1.83e11 - 2.70e11i)T^{2} \) |
| 89 | \( 1 + (1.69e5 - 7.71e5i)T + (-4.51e11 - 2.08e11i)T^{2} \) |
| 97 | \( 1 + (-1.42e6 + 9.64e5i)T + (3.08e11 - 7.73e11i)T^{2} \) |
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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
−13.02580396903947937898685662530, −11.39092648722880016383302449484, −10.64004095242776741133306282128, −10.53876617814802113466365221669, −9.183917601021547249236841687909, −7.49566503106718369491653363419, −4.73513090366247056201588688150, −3.93646941572441041755944167825, −2.41237913626229515419307968075, −0.05942118299399765262501772696,
1.61356285318745362349027305729, 5.30815218210106867742381652244, 5.80007540630520269729354747164, 7.52535299467922964068514748399, 8.160063440744229902876666486752, 8.868143233894852855715879498554, 11.41726538744578820507539586282, 12.50306835313968830044559452830, 13.46767620461895159982222944577, 14.61230325981254222590118665113
