L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.292 + 0.707i)11-s + 1.00·14-s − 1.00·16-s + (−0.292 + 0.707i)22-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (0.707 + 0.707i)28-s + (−0.292 + 0.707i)29-s + (−0.707 − 0.707i)32-s + (−0.707 + 0.292i)37-s + (0.292 + 0.707i)43-s + (−0.707 + 0.292i)44-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.292 + 0.707i)11-s + 1.00·14-s − 1.00·16-s + (−0.292 + 0.707i)22-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (0.707 + 0.707i)28-s + (−0.292 + 0.707i)29-s + (−0.707 − 0.707i)32-s + (−0.707 + 0.292i)37-s + (0.292 + 0.707i)43-s + (−0.707 + 0.292i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.796879012\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.796879012\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.311644417973603297272976831977, −8.541821384150125325796457518781, −7.73432988786425470515712801165, −7.09845294516708040140265300743, −6.51135740392107421976757165740, −5.31737962362532495786441592097, −4.77996484760534072971786166838, −3.95233729683041117742203099427, −3.01430681954288192439557345363, −1.63003276200212640078169806103,
1.21396122703845625330301437489, 2.38537325596200038892950584545, 3.21377672052894050813095894830, 4.27472878345829740229540383348, 5.07920313669870057411764089470, 5.77031409396254970914550213876, 6.56444144194694099992397944840, 7.57151314928219327473854439983, 8.813062340494490450085322662749, 8.981635027706545916795351829808