L(s) = 1 | + (0.0771 − 1.41i)2-s + (−0.945 + 2.59i)3-s + (−1.98 − 0.217i)4-s + (1.15 − 0.202i)5-s + (3.59 + 1.53i)6-s + (−2.48 + 4.29i)7-s + (−0.461 + 2.79i)8-s + (−3.56 − 2.98i)9-s + (−0.197 − 1.64i)10-s + (1.12 − 0.651i)11-s + (2.44 − 4.96i)12-s + (1.56 + 4.30i)13-s + (5.87 + 3.83i)14-s + (−0.561 + 3.18i)15-s + (3.90 + 0.866i)16-s + (2.55 − 2.14i)17-s + ⋯ |
L(s) = 1 | + (0.0545 − 0.998i)2-s + (−0.546 + 1.50i)3-s + (−0.994 − 0.108i)4-s + (0.514 − 0.0907i)5-s + (1.46 + 0.627i)6-s + (−0.937 + 1.62i)7-s + (−0.163 + 0.986i)8-s + (−1.18 − 0.996i)9-s + (−0.0625 − 0.518i)10-s + (0.340 − 0.196i)11-s + (0.706 − 1.43i)12-s + (0.434 + 1.19i)13-s + (1.57 + 1.02i)14-s + (−0.144 + 0.821i)15-s + (0.976 + 0.216i)16-s + (0.619 − 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.689785 + 0.438462i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.689785 + 0.438462i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0771 + 1.41i)T \) |
| 19 | \( 1 + (-0.369 + 4.34i)T \) |
good | 3 | \( 1 + (0.945 - 2.59i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-1.15 + 0.202i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (2.48 - 4.29i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.12 + 0.651i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.56 - 4.30i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.55 + 2.14i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.0598 - 0.339i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.903 - 1.07i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 4.49i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.14iT - 37T^{2} \) |
| 41 | \( 1 + (-8.84 - 3.22i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-6.63 + 1.17i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.689 + 0.578i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.26 - 0.223i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.06 - 7.23i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.91 - 0.513i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.95 + 2.33i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.96 - 11.1i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.39 + 0.507i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-8.75 - 3.18i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (11.1 + 6.46i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.44 + 0.889i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (2.82 - 2.36i)T + (16.8 - 95.5i)T^{2} \) |
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
−12.94328241559311808714859246027, −11.74588394012145423877395426057, −11.35480870865695582511603182640, −9.952804002701053230911635123686, −9.362896707400798568398943728505, −8.899050008463566749988849681797, −6.09375338333284584954855402496, −5.31062471918118573730478316214, −4.03286525641793394936859341010, −2.67683546586599988407837942104,
0.902935274516391621525712122833, 3.74924538977269867582065069391, 5.76235079823666212021366649287, 6.39162579756086891106893718813, 7.37052083703262288813702345544, 8.021572095798052925796217740611, 9.774752654971932506678409640826, 10.62929807140214226619270887253, 12.44195759954777889388645127077, 12.91714993444248860663846932328