L(s) = 1 | + (0.651 − 1.25i)2-s + (−1.52 + 0.818i)3-s + (−1.15 − 1.63i)4-s + (−0.995 − 0.532i)5-s + (0.0327 + 2.44i)6-s + (0.237 + 0.0473i)7-s + (−2.80 + 0.378i)8-s + (1.66 − 2.49i)9-s + (−1.31 + 0.902i)10-s + (−3.62 + 2.97i)11-s + (3.09 + 1.55i)12-s + (−2.36 + 1.26i)13-s + (0.214 − 0.267i)14-s + (1.95 − 0.00257i)15-s + (−1.35 + 3.76i)16-s + (−1.41 + 0.584i)17-s + ⋯ |
L(s) = 1 | + (0.460 − 0.887i)2-s + (−0.881 + 0.472i)3-s + (−0.575 − 0.817i)4-s + (−0.445 − 0.237i)5-s + (0.0133 + 0.999i)6-s + (0.0899 + 0.0178i)7-s + (−0.990 + 0.133i)8-s + (0.553 − 0.832i)9-s + (−0.416 + 0.285i)10-s + (−1.09 + 0.895i)11-s + (0.893 + 0.448i)12-s + (−0.655 + 0.350i)13-s + (0.0573 − 0.0715i)14-s + (0.504 − 0.000664i)15-s + (−0.337 + 0.941i)16-s + (−0.342 + 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0221911 + 0.0386067i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0221911 + 0.0386067i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.651 + 1.25i)T \) |
| 3 | \( 1 + (1.52 - 0.818i)T \) |
good | 5 | \( 1 + (0.995 + 0.532i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.237 - 0.0473i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (3.62 - 2.97i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (2.36 - 1.26i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (1.41 - 0.584i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.07 - 3.53i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (2.30 + 1.54i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (0.329 + 0.270i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (2.76 - 2.76i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.220 + 0.725i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (0.703 + 0.470i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.314 + 3.18i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-6.23 + 2.58i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-5.81 + 4.77i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (5.39 + 2.88i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (1.29 + 13.1i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (0.695 - 0.0684i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (5.49 + 1.09i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (1.31 + 6.59i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (2.49 - 6.02i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (4.21 - 13.8i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-8.98 - 13.4i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-3.38 - 3.38i)T + 97iT^{2} \) |
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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
−11.77892832611976735197301375555, −10.69300844703902946212030777470, −10.18460773810775250463308695763, −9.317227339753124049943642438282, −8.010400642749729617305238185894, −6.66036828436612478639903873650, −5.44375831149199211857470043390, −4.68657155275461600890272675597, −3.82082487655025519229413048819, −2.10068442338898599925828382469,
0.02687052565797944464357165496, 2.82967305682595673347908015170, 4.38059787604915513988320458473, 5.38895769988971823630564505304, 6.11491143643312664840963783837, 7.37002293190819638117163265181, 7.71049046696515957290011256394, 8.909770337925193822367269702463, 10.30750164688511722999449865221, 11.27780892486302837819948065389