L(s) = 1 | + (0.5 + 2.59i)7-s + (−1.5 + 0.866i)13-s + (3 − 1.73i)19-s + (2.5 + 4.33i)25-s + 1.73i·31-s + (−0.5 − 0.866i)37-s + (−6.5 + 11.2i)43-s + (−6.5 + 2.59i)49-s + 15.5i·61-s + 11·67-s + (−12 − 6.92i)73-s − 13·79-s + (−3 − 3.46i)91-s + (4.5 + 2.59i)97-s + (16.5 + 9.52i)103-s + ⋯ |
L(s) = 1 | + (0.188 + 0.981i)7-s + (−0.416 + 0.240i)13-s + (0.688 − 0.397i)19-s + (0.5 + 0.866i)25-s + 0.311i·31-s + (−0.0821 − 0.142i)37-s + (−0.991 + 1.71i)43-s + (−0.928 + 0.371i)49-s + 1.99i·61-s + 1.34·67-s + (−1.40 − 0.810i)73-s − 1.46·79-s + (−0.314 − 0.363i)91-s + (0.456 + 0.263i)97-s + (1.62 + 0.938i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.420761493\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.420761493\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.5 - 11.2i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5iT - 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (12 + 6.92i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.5 - 2.59i)T + (48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.178897769789374840228326136303, −8.578712919384978819319673265304, −7.68952909875811283598634950756, −6.94927096091884629168902746218, −6.04630490301077515619014444170, −5.24628324518402724872483446401, −4.59696857323131445477521214691, −3.32459527405826226300462766013, −2.53066720823162018393243113528, −1.36909543619551245611186742485,
0.50382477383887372135006834477, 1.78380373860135650601105111534, 3.04216336481633787324025528550, 3.93027957803695273354036887392, 4.78494597310027743368799352860, 5.58342178553037669370649361847, 6.64528674844189043892446630343, 7.25182491726101085399992354996, 8.019249613283833509541588126839, 8.694748899908788055745499889006