| L(s) = 1 | + (−0.699 + 1.22i)2-s + (0.881 − 0.471i)3-s + (−1.02 − 1.71i)4-s + (−0.339 − 0.413i)5-s + (−0.0376 + 1.41i)6-s + (−4.31 + 2.88i)7-s + (2.82 − 0.0519i)8-s + (0.555 − 0.831i)9-s + (0.744 − 0.127i)10-s + (−3.29 + 0.999i)11-s + (−1.71 − 1.03i)12-s + (2.77 + 2.27i)13-s + (−0.525 − 7.32i)14-s + (−0.493 − 0.204i)15-s + (−1.91 + 3.51i)16-s + (−4.63 + 1.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.494 + 0.869i)2-s + (0.509 − 0.272i)3-s + (−0.510 − 0.859i)4-s + (−0.151 − 0.184i)5-s + (−0.0153 + 0.577i)6-s + (−1.63 + 1.09i)7-s + (0.999 − 0.0183i)8-s + (0.185 − 0.277i)9-s + (0.235 − 0.0403i)10-s + (−0.993 + 0.301i)11-s + (−0.493 − 0.298i)12-s + (0.769 + 0.631i)13-s + (−0.140 − 1.95i)14-s + (−0.127 − 0.0528i)15-s + (−0.478 + 0.878i)16-s + (−1.12 + 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0322122 + 0.469163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0322122 + 0.469163i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.699 - 1.22i)T \) |
| 3 | \( 1 + (-0.881 + 0.471i)T \) |
| good | 5 | \( 1 + (0.339 + 0.413i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (4.31 - 2.88i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (3.29 - 0.999i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-2.77 - 2.27i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (4.63 - 1.91i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.220 - 2.23i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (5.50 - 1.09i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.91 - 6.31i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-6.96 - 6.96i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.66 + 0.262i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (0.485 + 2.44i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (5.65 + 3.02i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (2.20 + 5.31i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (2.24 + 7.40i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-7.81 + 6.41i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-5.37 - 10.0i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-2.12 - 3.97i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-1.78 - 2.66i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-0.963 - 0.644i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (3.81 - 9.20i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (10.4 - 1.02i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-14.4 - 2.86i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-9.74 - 9.74i)T + 97iT^{2} \) |
| 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
−11.95654569512039226829927321600, −10.39124319521548221853656491786, −9.778046797396802443001073659807, −8.593874428406616644797676304253, −8.493108871973169431347610255409, −6.88996701910288072677666869633, −6.38063212695423898905484271903, −5.28307995279052009191502265947, −3.73681735299958803575169279243, −2.18030241424846685910414682357,
0.32689864893982278771740072165, 2.65306924448595581744146270883, 3.46877638594566863498217857500, 4.43042658848670571677874936127, 6.27058595143128903255482441617, 7.44290610180761968994298297757, 8.231399122162883496172802769296, 9.380314561226210495138988800571, 10.01837856927383727950828563986, 10.69323624377392592739093493807
