| 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
−12.04568777430314322190153542772, −11.47169515941895321245205723289, −10.13342633868674056294821287659, −9.242152095412661640858647191849, −8.519400347482232913985882355025, −7.30075048549099554282571881387, −5.01584697380158848529626281626, −4.38117749521711095656633707353, −3.17081734580043377350698967655, −0.69217376080683220018837148271,
3.03519706156782225900984700141, 4.33712998380714510735074483328, 6.01378874257834466152570730695, 7.05394180514914228962562180951, 7.73011677390404523884687879200, 8.512106863325876988966101444025, 10.21985064323952414894400350062, 10.86235012631127348471399459713, 12.38216039495561878274664461249, 13.12501731945546747353637179339
