L(s) = 1 | + (0.0922 − 0.995i)2-s + (−0.982 − 0.183i)4-s + (0.149 + 0.526i)5-s + (−0.273 + 0.961i)8-s + (0.445 + 0.895i)9-s + (0.538 − 0.100i)10-s + (−0.243 + 0.857i)13-s + (0.932 + 0.361i)16-s + (0.932 + 0.361i)17-s + (0.932 − 0.361i)18-s + (−0.0505 − 0.544i)20-s + (0.595 − 0.368i)25-s + (0.831 + 0.322i)26-s + (−0.876 − 0.163i)29-s + (0.445 − 0.895i)32-s + ⋯ |
L(s) = 1 | + (0.0922 − 0.995i)2-s + (−0.982 − 0.183i)4-s + (0.149 + 0.526i)5-s + (−0.273 + 0.961i)8-s + (0.445 + 0.895i)9-s + (0.538 − 0.100i)10-s + (−0.243 + 0.857i)13-s + (0.932 + 0.361i)16-s + (0.932 + 0.361i)17-s + (0.932 − 0.361i)18-s + (−0.0505 − 0.544i)20-s + (0.595 − 0.368i)25-s + (0.831 + 0.322i)26-s + (−0.876 − 0.163i)29-s + (0.445 − 0.895i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.028272758\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028272758\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0922 + 0.995i)T \) |
| 17 | \( 1 + (-0.932 - 0.361i)T \) |
good | 3 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 5 | \( 1 + (-0.149 - 0.526i)T + (-0.850 + 0.526i)T^{2} \) |
| 7 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 11 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 13 | \( 1 + (0.243 - 0.857i)T + (-0.850 - 0.526i)T^{2} \) |
| 19 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 23 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 29 | \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \) |
| 31 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 37 | \( 1 + (-0.136 + 0.124i)T + (0.0922 - 0.995i)T^{2} \) |
| 41 | \( 1 + (-0.465 - 0.288i)T + (0.445 + 0.895i)T^{2} \) |
| 43 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 47 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 53 | \( 1 + (0.537 + 1.07i)T + (-0.602 + 0.798i)T^{2} \) |
| 59 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 61 | \( 1 + (-1.02 + 1.35i)T + (-0.273 - 0.961i)T^{2} \) |
| 67 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 71 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 73 | \( 1 + (-0.831 - 0.322i)T + (0.739 + 0.673i)T^{2} \) |
| 79 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 83 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 89 | \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \) |
| 97 | \( 1 + (0.876 + 1.75i)T + (-0.602 + 0.798i)T^{2} \) |
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
−9.953477164719401041686363948857, −9.522795557698738440267417333839, −8.391470006701298987495516336703, −7.64281182875198289279389161523, −6.58692353749896977009609223009, −5.47634773724758441359996637950, −4.62252929534954542351558152884, −3.68261091018087324305168542351, −2.57070085652996480089114490547, −1.62158317312793800906900972005,
1.03519128124228648072767871759, 3.13913344029312502229628573526, 4.08483513147475854573868700891, 5.15945654741979196312564627919, 5.73739918944325260986483344859, 6.76799508878697614185479077450, 7.50681395052233405952337977921, 8.291335683936464543487307269437, 9.217450227575463275735028636677, 9.677530774044022371213489467797