L(s) = 1 | + (−0.120 + 0.992i)3-s + (0.845 − 0.534i)4-s + (0.948 + 0.316i)7-s + (−0.970 − 0.239i)9-s + (0.428 + 0.903i)12-s + (0.5 − 0.866i)13-s + (0.428 − 0.903i)16-s − 1.92i·19-s + (−0.428 + 0.903i)21-s + (0.632 − 0.774i)25-s + (0.354 − 0.935i)27-s + (0.970 − 0.239i)28-s + (−0.149 − 0.917i)31-s + (−0.948 + 0.316i)36-s + (0.101 + 0.625i)37-s + ⋯ |
L(s) = 1 | + (−0.120 + 0.992i)3-s + (0.845 − 0.534i)4-s + (0.948 + 0.316i)7-s + (−0.970 − 0.239i)9-s + (0.428 + 0.903i)12-s + (0.5 − 0.866i)13-s + (0.428 − 0.903i)16-s − 1.92i·19-s + (−0.428 + 0.903i)21-s + (0.632 − 0.774i)25-s + (0.354 − 0.935i)27-s + (0.970 − 0.239i)28-s + (−0.149 − 0.917i)31-s + (−0.948 + 0.316i)36-s + (0.101 + 0.625i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.666223885\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.666223885\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.120 - 0.992i)T \) |
| 7 | \( 1 + (-0.948 - 0.316i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.845 + 0.534i)T^{2} \) |
| 5 | \( 1 + (-0.632 + 0.774i)T^{2} \) |
| 11 | \( 1 + (0.885 - 0.464i)T^{2} \) |
| 17 | \( 1 + (-0.799 + 0.600i)T^{2} \) |
| 19 | \( 1 + 1.92iT - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.0402 + 0.999i)T^{2} \) |
| 31 | \( 1 + (0.149 + 0.917i)T + (-0.948 + 0.316i)T^{2} \) |
| 37 | \( 1 + (-0.101 - 0.625i)T + (-0.948 + 0.316i)T^{2} \) |
| 41 | \( 1 + (0.278 + 0.960i)T^{2} \) |
| 43 | \( 1 + (1.19 - 1.46i)T + (-0.200 - 0.979i)T^{2} \) |
| 47 | \( 1 + (-0.996 - 0.0804i)T^{2} \) |
| 53 | \( 1 + (0.919 + 0.391i)T^{2} \) |
| 59 | \( 1 + (-0.632 + 0.774i)T^{2} \) |
| 61 | \( 1 + (1.26 - 1.12i)T + (0.120 - 0.992i)T^{2} \) |
| 67 | \( 1 + (-0.869 - 1.65i)T + (-0.568 + 0.822i)T^{2} \) |
| 71 | \( 1 + (0.692 - 0.721i)T^{2} \) |
| 73 | \( 1 + (1.30 - 1.25i)T + (0.0402 - 0.999i)T^{2} \) |
| 79 | \( 1 + (-0.00970 + 0.240i)T + (-0.996 - 0.0804i)T^{2} \) |
| 83 | \( 1 + (-0.970 + 0.239i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.56 + 0.742i)T + (0.632 + 0.774i)T^{2} \) |
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
−8.727137746718792339391586201364, −8.175676248726794992426004991529, −7.23307458869576090059976821721, −6.34794759435236482859979718142, −5.66331420212818014987095002264, −4.96735720197176233720401144382, −4.36269950662993969655358619602, −3.01845584447451220058362237822, −2.50221431495285931156813314182, −1.06475335873097170054927313640,
1.61283309155761557172944988858, 1.79427568790627461487621470649, 3.17107142945812954185343984050, 3.91403574943778683189965622271, 5.12222192522003649642108331549, 5.92858990460274020511070805603, 6.69659553741038756144645434524, 7.23965359843709753387492946026, 7.972034728507066787306825398538, 8.378406476425640878663609680570