L(s) = 1 | + (0.870 − 0.492i)2-s + (0.515 − 0.856i)4-s + (1.37 − 1.44i)5-s + (0.0270 − 0.999i)8-s + (−0.492 + 0.870i)9-s + (0.481 − 1.93i)10-s + (0.154 + 0.749i)13-s + (−0.468 − 0.883i)16-s + (−0.916 − 1.23i)17-s + i·18-s + (−0.533 − 1.92i)20-s + (−0.160 − 2.96i)25-s + (0.503 + 0.576i)26-s + (−1.68 + 1.07i)29-s + (−0.842 − 0.538i)32-s + ⋯ |
L(s) = 1 | + (0.870 − 0.492i)2-s + (0.515 − 0.856i)4-s + (1.37 − 1.44i)5-s + (0.0270 − 0.999i)8-s + (−0.492 + 0.870i)9-s + (0.481 − 1.93i)10-s + (0.154 + 0.749i)13-s + (−0.468 − 0.883i)16-s + (−0.916 − 1.23i)17-s + i·18-s + (−0.533 − 1.92i)20-s + (−0.160 − 2.96i)25-s + (0.503 + 0.576i)26-s + (−1.68 + 1.07i)29-s + (−0.842 − 0.538i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.579738086\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.579738086\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.870 + 0.492i)T \) |
| 929 | \( 1 - iT \) |
good | 3 | \( 1 + (0.492 - 0.870i)T^{2} \) |
| 5 | \( 1 + (-1.37 + 1.44i)T + (-0.0541 - 0.998i)T^{2} \) |
| 7 | \( 1 + (0.779 + 0.626i)T^{2} \) |
| 11 | \( 1 + (-0.687 - 0.725i)T^{2} \) |
| 13 | \( 1 + (-0.154 - 0.749i)T + (-0.918 + 0.395i)T^{2} \) |
| 17 | \( 1 + (0.916 + 1.23i)T + (-0.293 + 0.955i)T^{2} \) |
| 19 | \( 1 + (0.725 + 0.687i)T^{2} \) |
| 23 | \( 1 + (0.856 - 0.515i)T^{2} \) |
| 29 | \( 1 + (1.68 - 1.07i)T + (0.419 - 0.907i)T^{2} \) |
| 31 | \( 1 + (-0.842 - 0.538i)T^{2} \) |
| 37 | \( 1 + (-1.55 - 1.05i)T + (0.370 + 0.928i)T^{2} \) |
| 41 | \( 1 + (-1.74 - 0.166i)T + (0.982 + 0.188i)T^{2} \) |
| 43 | \( 1 + (0.492 - 0.870i)T^{2} \) |
| 47 | \( 1 + (0.214 + 0.976i)T^{2} \) |
| 53 | \( 1 + (-1.20 + 0.0655i)T + (0.994 - 0.108i)T^{2} \) |
| 59 | \( 1 + (0.135 + 0.990i)T^{2} \) |
| 61 | \( 1 + (-0.495 - 1.15i)T + (-0.687 + 0.725i)T^{2} \) |
| 67 | \( 1 + (-0.188 - 0.982i)T^{2} \) |
| 71 | \( 1 + (0.108 + 0.994i)T^{2} \) |
| 73 | \( 1 + (1.27 - 0.0861i)T + (0.990 - 0.135i)T^{2} \) |
| 79 | \( 1 + (0.744 + 0.667i)T^{2} \) |
| 83 | \( 1 + (-0.214 - 0.976i)T^{2} \) |
| 89 | \( 1 + (-1.31 - 0.893i)T + (0.370 + 0.928i)T^{2} \) |
| 97 | \( 1 + (-0.296 - 1.97i)T + (-0.955 + 0.293i)T^{2} \) |
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
−8.870329602834962319496952062471, −7.72232105091247771332308977134, −6.71809891152458824023512949508, −5.93714638486056176407128728755, −5.32007756586648689491854425939, −4.79845526157115885417364304240, −4.13563214717539372892509941561, −2.63154789140371984203911430714, −2.10664209532665092483185111505, −1.13826949905962452901603906269,
1.99219921345663555891713203478, 2.64717261854598395719623588819, 3.47532596916902966164612547184, 4.20523212983189231386142841105, 5.68418918820815923427015616194, 5.90431392969490277337520612048, 6.38125681206848660562263342530, 7.23037743821425360973029109214, 7.88105071345513523349410309163, 8.963655878648578112092556006101