| L(s) = 1 | + (−0.347 − 1.96i)2-s + (−0.666 + 3.77i)3-s + (−3.75 + 1.36i)4-s + (−4.75 − 3.99i)5-s + 7.67·6-s + (19.2 + 16.1i)7-s + (4 + 6.92i)8-s + (11.5 + 4.19i)9-s + (−6.20 + 10.7i)10-s + (30.0 + 51.9i)11-s + (−2.66 − 15.1i)12-s + (−44.7 + 16.2i)13-s + (25.1 − 43.4i)14-s + (18.2 − 15.3i)15-s + (12.2 − 10.2i)16-s + (25.2 + 9.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.128 + 0.727i)3-s + (−0.469 + 0.171i)4-s + (−0.425 − 0.356i)5-s + 0.522·6-s + (1.03 + 0.871i)7-s + (0.176 + 0.306i)8-s + (0.427 + 0.155i)9-s + (−0.196 + 0.340i)10-s + (0.822 + 1.42i)11-s + (−0.0641 − 0.363i)12-s + (−0.954 + 0.347i)13-s + (0.479 − 0.830i)14-s + (0.314 − 0.263i)15-s + (0.191 − 0.160i)16-s + (0.360 + 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.22619 + 0.401281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22619 + 0.401281i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 + 1.96i)T \) |
| 37 | \( 1 + (97.9 - 202. i)T \) |
| good | 3 | \( 1 + (0.666 - 3.77i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (4.75 + 3.99i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-19.2 - 16.1i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (-30.0 - 51.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (44.7 - 16.2i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-25.2 - 9.19i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 19 | \( 1 + (0.729 - 4.13i)T + (-6.44e3 - 2.34e3i)T^{2} \) |
| 23 | \( 1 + (-70.3 + 121. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-32.9 - 57.0i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 298.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-132. + 48.3i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 - 503.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-58.4 + 101. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-121. + 101. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-284. + 238. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-274. + 100. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (769. + 646. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-35.2 + 199. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (58.7 + 49.3i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-771. - 280. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-244. + 205. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-473. + 820. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.52338056425346520265631363443, −12.52421322299688002280021482829, −12.08739315169848600491053282007, −10.86588152711756802175978961512, −9.722184683500743119954979470855, −8.801699044008749759805180316012, −7.36230827837734328225088812028, −5.00004919637266690984951900139, −4.28007904002439636411719734770, −1.95598397228148885354934216079,
1.00750353405122609447784960711, 3.90842420851731071994001659167, 5.64482300983857441853249766961, 7.25732972601774286248696120477, 7.61343422084196780543852426512, 9.158845251030634673714126672405, 10.77387496819277736763625861630, 11.72062998085075396458952338288, 13.10492306690032829753437257663, 14.13483700911273503647958879572
