| L(s) = 1 | + (0.0153 + 1.41i)2-s + (0.335 − 0.734i)3-s + (−1.99 + 0.0433i)4-s + (−2.69 + 3.11i)5-s + (1.04 + 0.463i)6-s + (−1.00 + 0.645i)7-s + (−0.0919 − 2.82i)8-s + (1.53 + 1.77i)9-s + (−4.44 − 3.76i)10-s + (−2.79 − 0.402i)11-s + (−0.638 + 1.48i)12-s + (0.385 − 0.599i)13-s + (−0.927 − 1.40i)14-s + (1.38 + 3.02i)15-s + (3.99 − 0.173i)16-s + (−1.46 + 4.98i)17-s + ⋯ |
| L(s) = 1 | + (0.0108 + 0.999i)2-s + (0.193 − 0.424i)3-s + (−0.999 + 0.0216i)4-s + (−1.20 + 1.39i)5-s + (0.426 + 0.189i)6-s + (−0.379 + 0.243i)7-s + (−0.0324 − 0.999i)8-s + (0.512 + 0.591i)9-s + (−1.40 − 1.19i)10-s + (−0.843 − 0.121i)11-s + (−0.184 + 0.428i)12-s + (0.106 − 0.166i)13-s + (−0.247 − 0.376i)14-s + (0.356 + 0.780i)15-s + (0.999 − 0.0433i)16-s + (−0.355 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.139098 + 0.736837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.139098 + 0.736837i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0153 - 1.41i)T \) |
| 23 | \( 1 + (-4.34 + 2.03i)T \) |
| good | 3 | \( 1 + (-0.335 + 0.734i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (2.69 - 3.11i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (1.00 - 0.645i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (2.79 + 0.402i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.385 + 0.599i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.46 - 4.98i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.987 - 3.36i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (1.87 - 6.40i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-8.74 + 3.99i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (3.47 + 4.00i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.75 + 2.02i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.28 - 1.04i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.53iT - 47T^{2} \) |
| 53 | \( 1 + (-5.97 + 3.84i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.95 + 1.90i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.52 - 5.52i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (7.59 - 1.09i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (6.66 - 0.958i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (1.06 - 0.312i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (5.61 + 3.61i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (2.25 - 1.94i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.20 - 0.551i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.77 - 2.40i)T + (13.8 + 96.0i)T^{2} \) |
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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
−13.12501731945546747353637179339, −12.38216039495561878274664461249, −10.86235012631127348471399459713, −10.21985064323952414894400350062, −8.512106863325876988966101444025, −7.73011677390404523884687879200, −7.05394180514914228962562180951, −6.01378874257834466152570730695, −4.33712998380714510735074483328, −3.03519706156782225900984700141,
0.69217376080683220018837148271, 3.17081734580043377350698967655, 4.38117749521711095656633707353, 5.01584697380158848529626281626, 7.30075048549099554282571881387, 8.519400347482232913985882355025, 9.242152095412661640858647191849, 10.13342633868674056294821287659, 11.47169515941895321245205723289, 12.04568777430314322190153542772
