L(s) = 1 | + (−0.961 + 0.273i)2-s + (0.850 − 0.526i)4-s + (−0.0922 + 1.99i)5-s + (−0.673 + 0.739i)8-s + (0.183 + 0.982i)9-s + (−0.457 − 1.94i)10-s + (−0.271 − 0.247i)13-s + (0.445 − 0.895i)16-s + (−0.445 + 0.895i)17-s + (−0.445 − 0.895i)18-s + (0.972 + 1.74i)20-s + (−2.97 − 0.276i)25-s + (0.328 + 0.163i)26-s + (0.377 − 1.60i)29-s + (−0.183 + 0.982i)32-s + ⋯ |
L(s) = 1 | + (−0.961 + 0.273i)2-s + (0.850 − 0.526i)4-s + (−0.0922 + 1.99i)5-s + (−0.673 + 0.739i)8-s + (0.183 + 0.982i)9-s + (−0.457 − 1.94i)10-s + (−0.271 − 0.247i)13-s + (0.445 − 0.895i)16-s + (−0.445 + 0.895i)17-s + (−0.445 − 0.895i)18-s + (0.972 + 1.74i)20-s + (−2.97 − 0.276i)25-s + (0.328 + 0.163i)26-s + (0.377 − 1.60i)29-s + (−0.183 + 0.982i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5910302806\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5910302806\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.961 - 0.273i)T \) |
| 17 | \( 1 + (0.445 - 0.895i)T \) |
good | 3 | \( 1 + (-0.183 - 0.982i)T^{2} \) |
| 5 | \( 1 + (0.0922 - 1.99i)T + (-0.995 - 0.0922i)T^{2} \) |
| 7 | \( 1 + (-0.361 - 0.932i)T^{2} \) |
| 11 | \( 1 + (-0.895 - 0.445i)T^{2} \) |
| 13 | \( 1 + (0.271 + 0.247i)T + (0.0922 + 0.995i)T^{2} \) |
| 19 | \( 1 + (-0.850 - 0.526i)T^{2} \) |
| 23 | \( 1 + (-0.361 - 0.932i)T^{2} \) |
| 29 | \( 1 + (-0.377 + 1.60i)T + (-0.895 - 0.445i)T^{2} \) |
| 31 | \( 1 + (-0.995 + 0.0922i)T^{2} \) |
| 37 | \( 1 + (0.134 + 0.963i)T + (-0.961 + 0.273i)T^{2} \) |
| 41 | \( 1 + (0.0590 - 0.0710i)T + (-0.183 - 0.982i)T^{2} \) |
| 43 | \( 1 + (-0.602 + 0.798i)T^{2} \) |
| 47 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 53 | \( 1 + (-0.132 - 0.710i)T + (-0.932 + 0.361i)T^{2} \) |
| 59 | \( 1 + (0.739 + 0.673i)T^{2} \) |
| 61 | \( 1 + (-1.16 - 0.516i)T + (0.673 + 0.739i)T^{2} \) |
| 67 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 71 | \( 1 + (0.361 + 0.932i)T^{2} \) |
| 73 | \( 1 + (-0.359 - 1.07i)T + (-0.798 + 0.602i)T^{2} \) |
| 79 | \( 1 + (-0.526 + 0.850i)T^{2} \) |
| 83 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
| 89 | \( 1 + (1.42 - 1.29i)T + (0.0922 - 0.995i)T^{2} \) |
| 97 | \( 1 + (1.10 - 1.60i)T + (-0.361 - 0.932i)T^{2} \) |
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
−10.19654326819589028865588584782, −9.813329249594524013781989533644, −8.475746569550957035104538016736, −7.70686198476924681754422369795, −7.18969871504468193523925094406, −6.35173046435067563450922290202, −5.65827885123246235803309280798, −4.06195180476313779918231823928, −2.73407228026501518191925889141, −2.11235309736635328512070535690,
0.71297059867034199720794574493, 1.79857956654399455668092766484, 3.37656283740855865314063876012, 4.47121627280449241303164359610, 5.35557179032733201960467522213, 6.55215079085436351405773944098, 7.38814475387878325932612616539, 8.478515367490402471691811393708, 8.840728682437282077512493371611, 9.511298904193538308274607214665