L(s) = 1 | + (1.30 − 0.124i)2-s + (0.702 − 0.135i)4-s + (−0.357 + 0.105i)8-s + (0.0475 − 0.998i)9-s + (1.91 + 0.182i)11-s + (−1.11 + 0.447i)16-s + (−0.0623 − 1.30i)18-s + 2.51·22-s + (−0.888 − 0.458i)23-s + (0.580 + 0.814i)25-s + (−0.544 + 0.627i)29-s + (−1.06 + 0.551i)32-s + (−0.101 − 0.708i)36-s + (−0.0135 + 0.284i)37-s + (−1.61 − 0.474i)43-s + (1.36 − 0.130i)44-s + ⋯ |
L(s) = 1 | + (1.30 − 0.124i)2-s + (0.702 − 0.135i)4-s + (−0.357 + 0.105i)8-s + (0.0475 − 0.998i)9-s + (1.91 + 0.182i)11-s + (−1.11 + 0.447i)16-s + (−0.0623 − 1.30i)18-s + 2.51·22-s + (−0.888 − 0.458i)23-s + (0.580 + 0.814i)25-s + (−0.544 + 0.627i)29-s + (−1.06 + 0.551i)32-s + (−0.101 − 0.708i)36-s + (−0.0135 + 0.284i)37-s + (−1.61 − 0.474i)43-s + (1.36 − 0.130i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.974388507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.974388507\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 + (0.888 + 0.458i)T \) |
good | 2 | \( 1 + (-1.30 + 0.124i)T + (0.981 - 0.189i)T^{2} \) |
| 3 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 5 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 11 | \( 1 + (-1.91 - 0.182i)T + (0.981 + 0.189i)T^{2} \) |
| 13 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 19 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 29 | \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \) |
| 31 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 37 | \( 1 + (0.0135 - 0.284i)T + (-0.995 - 0.0950i)T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.653 + 0.513i)T + (0.235 + 0.971i)T^{2} \) |
| 59 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 61 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 67 | \( 1 + (0.759 + 1.06i)T + (-0.327 + 0.945i)T^{2} \) |
| 71 | \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 79 | \( 1 + (0.653 - 0.513i)T + (0.235 - 0.971i)T^{2} \) |
| 83 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 97 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
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
−9.914997876149488453405149702320, −9.160049533475300004430028032620, −8.559939287492618565042402639086, −7.00543036167023001064590851339, −6.50629044153510734780375990130, −5.70986737597071232596647571109, −4.61523165659494800618062044406, −3.83159259425709233753262073087, −3.22206699052356521857904845351, −1.62494510493391241916724470700,
1.82028354144442201061651757711, 3.16631025946390366882105303957, 4.11618816041872421846124387341, 4.71646718554379717331674140035, 5.81925956307113370636081061902, 6.40447775763384302392803298112, 7.31684493131889358635649332595, 8.406961063222045039267603691894, 9.239134287394789347877519023417, 10.07925493703224664523971706857