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 | \( 1 + (-473. - 820. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.13483700911273503647958879572, −13.10492306690032829753437257663, −11.72062998085075396458952338288, −10.77387496819277736763625861630, −9.158845251030634673714126672405, −7.61343422084196780543852426512, −7.25732972601774286248696120477, −5.64482300983857441853249766961, −3.90842420851731071994001659167, −1.00750353405122609447784960711,
1.95598397228148885354934216079, 4.28007904002439636411719734770, 5.00004919637266690984951900139, 7.36230827837734328225088812028, 8.801699044008749759805180316012, 9.722184683500743119954979470855, 10.86588152711756802175978961512, 12.08739315169848600491053282007, 12.52421322299688002280021482829, 14.52338056425346520265631363443