L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s + 0.999·10-s + (−0.499 − 0.866i)14-s + (0.5 − 0.866i)16-s − 19-s + (−0.5 − 0.866i)31-s + 0.999·35-s + (−0.5 + 0.866i)38-s + (0.500 + 0.866i)40-s + (0.5 + 0.866i)41-s + (−1 + 1.73i)47-s + (0.499 − 0.866i)56-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s + 0.999·10-s + (−0.499 − 0.866i)14-s + (0.5 − 0.866i)16-s − 19-s + (−0.5 − 0.866i)31-s + 0.999·35-s + (−0.5 + 0.866i)38-s + (0.500 + 0.866i)40-s + (0.5 + 0.866i)41-s + (−1 + 1.73i)47-s + (0.499 − 0.866i)56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.996083960\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.996083960\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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
−9.221495651712928731966551759378, −8.042432713449573657439550777471, −7.53193451433867613237083038236, −6.69488411466950172169269878839, −5.95243118436079434648780350514, −4.67652363777713945213208288333, −4.17203802612530014683683916551, −3.17678956610731496496379409508, −2.40667148361447175662672789022, −1.45679933328763767227800057567,
1.48504771842395592138719014485, 2.29532859755532386104098634165, 3.84794952580968565310933986284, 4.86240205450708418047270847085, 5.28884184663364560134744754457, 5.93748109579216907775557599478, 6.73599545411860166845314853552, 7.54173130094459759973046988136, 8.616701691119841433402697401360, 8.745969632896061782863364343870