| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.0553 − 1.73i)3-s + (−0.939 + 0.342i)4-s + (0.882 − 2.42i)5-s + (−1.69 + 0.355i)6-s + (−1.58 + 2.74i)7-s + (0.5 + 0.866i)8-s + (−2.99 + 0.191i)9-s + (−2.54 − 0.448i)10-s + (2.16 − 1.25i)11-s + (0.644 + 1.60i)12-s + (2.71 − 3.24i)13-s + (2.97 + 1.08i)14-s + (−4.24 − 1.39i)15-s + (0.766 − 0.642i)16-s + (−1.32 + 0.233i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.0319 − 0.999i)3-s + (−0.469 + 0.171i)4-s + (0.394 − 1.08i)5-s + (−0.692 + 0.144i)6-s + (−0.598 + 1.03i)7-s + (0.176 + 0.306i)8-s + (−0.997 + 0.0638i)9-s + (−0.803 − 0.141i)10-s + (0.653 − 0.377i)11-s + (0.185 + 0.464i)12-s + (0.754 − 0.898i)13-s + (0.795 + 0.289i)14-s + (−1.09 − 0.359i)15-s + (0.191 − 0.160i)16-s + (−0.320 + 0.0565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.467984 - 0.820436i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.467984 - 0.820436i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.0553 + 1.73i)T \) |
| 19 | \( 1 + (-3.14 - 3.01i)T \) |
| good | 5 | \( 1 + (-0.882 + 2.42i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.58 - 2.74i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.16 + 1.25i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.71 + 3.24i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.32 - 0.233i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.30 - 3.58i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.32 + 7.49i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-6.89 - 3.97i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.10iT - 37T^{2} \) |
| 41 | \( 1 + (4.95 - 4.16i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (11.7 + 4.27i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (6.16 + 1.08i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.46 + 1.26i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.54 - 8.75i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (0.133 - 0.0485i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (4.48 + 0.791i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (8.59 + 3.12i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (1.67 - 1.40i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.41 - 7.64i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (12.3 + 7.11i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-12.7 - 10.6i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.538 + 0.0949i)T + (91.1 - 33.1i)T^{2} \) |
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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
−13.25344761853728693417516119854, −12.12332704479552032515041531310, −11.67241860314474014857482985384, −9.951551411675276440184342894607, −8.834768417426439600105273640380, −8.220852541580195392015481680472, −6.29406986463138430768377161935, −5.33037507592124389968303135061, −3.12506621034275983850418507779, −1.33964862873533353556297718732,
3.37288811449141544272358512240, 4.62493903445833057641154411904, 6.40285839495024863312887549580, 6.96137578821887347878257558851, 8.755243760763336598851563294190, 9.778907939202467104685595414537, 10.52116882501486052384002199281, 11.52100418535152963633926255671, 13.42469319451217603352215894409, 14.16726128166956545024856805643
