| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.909 + 0.415i)3-s + (−0.959 + 0.281i)4-s + (−1.10 + 1.27i)5-s + (0.281 − 0.959i)6-s + (1.52 + 2.15i)7-s + (0.415 + 0.909i)8-s + (0.654 + 0.755i)9-s + (1.41 + 0.910i)10-s + (−3.28 − 0.472i)11-s + (−0.989 − 0.142i)12-s + (−0.697 + 1.08i)13-s + (1.92 − 1.82i)14-s + (−1.53 + 0.699i)15-s + (0.841 − 0.540i)16-s + (−0.711 − 0.208i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.525 + 0.239i)3-s + (−0.479 + 0.140i)4-s + (−0.493 + 0.568i)5-s + (0.115 − 0.391i)6-s + (0.577 + 0.816i)7-s + (0.146 + 0.321i)8-s + (0.218 + 0.251i)9-s + (0.447 + 0.287i)10-s + (−0.990 − 0.142i)11-s + (−0.285 − 0.0410i)12-s + (−0.193 + 0.300i)13-s + (0.513 − 0.486i)14-s + (−0.395 + 0.180i)15-s + (0.210 − 0.135i)16-s + (−0.172 − 0.0506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.614714 + 0.730773i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.614714 + 0.730773i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (-1.52 - 2.15i)T \) |
| 23 | \( 1 + (-4.41 + 1.86i)T \) |
| good | 5 | \( 1 + (1.10 - 1.27i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (3.28 + 0.472i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.697 - 1.08i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.711 + 0.208i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (6.49 - 1.90i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (5.76 + 1.69i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (8.46 - 3.86i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 1.13i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.0528 - 0.0457i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.79 - 2.19i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 0.942iT - 47T^{2} \) |
| 53 | \( 1 + (2.45 + 3.82i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (4.80 - 7.48i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.39 - 11.8i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-4.18 + 0.602i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.157 - 1.09i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.11 - 7.20i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-5.84 + 9.09i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.21 - 6.01i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.54 + 5.58i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-4.56 + 5.26i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.50896752241378136631333489595, −9.306098114502032663482700284661, −8.738401530614341651451525521735, −7.926703074724580988587785803250, −7.14795619554314204196756378541, −5.74925096146926998348336393571, −4.79924210672636331155810123991, −3.80054476877597241503518483834, −2.76437478090829897728697096936, −1.96430696168356510244939620297,
0.40956266883293658045592452993, 2.07160961273382319943798587320, 3.65351475657398018674098399328, 4.56944829384518569202600278281, 5.31594908599585646715236161119, 6.61497641613759681355955113914, 7.53648241636195101066781099129, 7.928974824237977130729583555785, 8.740632682253608207385641520297, 9.517460098184839194166008601407
