L(s) = 1 | + (−1.50 − 1.26i)3-s + (0.500 + 2.83i)9-s + (1.32 + 0.766i)11-s + (−0.173 + 0.984i)17-s + (−0.342 + 0.939i)19-s + (0.766 − 0.642i)25-s + (1.85 − 3.20i)27-s + (−1.03 − 2.83i)33-s + (0.439 − 0.524i)41-s + (0.342 + 0.939i)43-s + (0.5 − 0.866i)49-s + (1.50 − 1.26i)51-s + (1.70 − 0.984i)57-s + (−0.223 + 1.26i)59-s + (−0.118 − 0.673i)67-s + ⋯ |
L(s) = 1 | + (−1.50 − 1.26i)3-s + (0.500 + 2.83i)9-s + (1.32 + 0.766i)11-s + (−0.173 + 0.984i)17-s + (−0.342 + 0.939i)19-s + (0.766 − 0.642i)25-s + (1.85 − 3.20i)27-s + (−1.03 − 2.83i)33-s + (0.439 − 0.524i)41-s + (0.342 + 0.939i)43-s + (0.5 − 0.866i)49-s + (1.50 − 1.26i)51-s + (1.70 − 0.984i)57-s + (−0.223 + 1.26i)59-s + (−0.118 − 0.673i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6694147326\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6694147326\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.342 - 0.939i)T \) |
good | 3 | \( 1 + (1.50 + 1.26i)T + (0.173 + 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.342 - 0.939i)T + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.223 - 1.26i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.118 + 0.673i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \) |
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
−10.27641339167269573519435032129, −8.998993558701019896141797040433, −7.979884364153337119762600971611, −7.24139452214718781095818405374, −6.38263135896529070721597018102, −6.08400663843026785017736672166, −4.92938352888714180568780801509, −4.05527060704066262215472225775, −2.12104111770290365155101052704, −1.23910926268359474326035321563,
0.874569412338624668585024776104, 3.16764822558467111678555864723, 4.12457433277935380765076252523, 4.84529431911140568271770440989, 5.68425049102192360720198570067, 6.47199588614834529112290557857, 7.10953149754410534283276758894, 8.857736404653668686460676085550, 9.215085877188752380496596817711, 10.02036443937041995306462382893