| L(s) = 1 | + (1.92 + 1.51i)2-s + (−0.261 + 0.367i)3-s + (0.941 + 3.88i)4-s + (−1.46 + 0.281i)5-s + (−1.06 + 0.311i)6-s + (−2.01 − 1.70i)7-s + (−2.02 + 4.43i)8-s + (0.914 + 2.64i)9-s + (−3.24 − 1.67i)10-s + (4.03 − 3.17i)11-s + (−1.67 − 0.670i)12-s + (5.59 − 3.59i)13-s + (−1.30 − 6.34i)14-s + (0.279 − 0.611i)15-s + (−3.52 + 1.81i)16-s + (−1.75 − 1.66i)17-s + ⋯ |
| L(s) = 1 | + (1.36 + 1.07i)2-s + (−0.151 + 0.212i)3-s + (0.470 + 1.94i)4-s + (−0.654 + 0.126i)5-s + (−0.432 + 0.127i)6-s + (−0.763 − 0.645i)7-s + (−0.716 + 1.56i)8-s + (0.304 + 0.880i)9-s + (−1.02 − 0.528i)10-s + (1.21 − 0.956i)11-s + (−0.483 − 0.193i)12-s + (1.55 − 0.997i)13-s + (−0.347 − 1.69i)14-s + (0.0721 − 0.157i)15-s + (−0.880 + 0.453i)16-s + (−0.424 − 0.404i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28752 + 1.42722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28752 + 1.42722i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.01 + 1.70i)T \) |
| 23 | \( 1 + (4.13 + 2.43i)T \) |
| good | 2 | \( 1 + (-1.92 - 1.51i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (0.261 - 0.367i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (1.46 - 0.281i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-4.03 + 3.17i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-5.59 + 3.59i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.75 + 1.66i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (4.76 - 4.54i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (3.87 - 1.13i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.22 - 0.116i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (0.215 + 0.622i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (0.187 + 0.216i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.936 + 2.04i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-3.09 + 5.35i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.199 - 4.18i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (0.755 + 0.389i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-5.51 - 7.74i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-2.12 + 0.852i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.78 + 12.3i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.738 - 3.04i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.00251 + 0.0528i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (4.18 - 4.82i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (5.34 + 0.510i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-9.58 - 11.0i)T + (-13.8 + 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.44183973183833374569314884003, −12.54595552509602636052692813132, −11.35270031566691023831846959655, −10.42577295340623079782697730883, −8.558764345204106602572228070240, −7.64169060800788805479060554527, −6.46355786833951885982459230387, −5.74517383353352792415677352528, −4.03204978482297457595226411379, −3.65086791590542915551626357582,
1.84293384230217039764859118603, 3.76378036614209192194091357211, 4.20672916733194856547963347659, 6.10120930579645407258255972457, 6.67803170338506635062232091846, 8.858428822026022097297610855904, 9.754805528894548042556769534892, 11.29048129619331070588223219647, 11.72893180805479263932604830396, 12.59456610394135128107431358334
