L(s) = 1 | + (1 − 1.73i)5-s + (2 + 1.73i)7-s + (−1 − 1.73i)11-s + (1.5 + 2.59i)13-s + (−4 + 6.92i)17-s + (0.5 + 0.866i)19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + (−2 + 3.46i)29-s + 3·31-s + (5 − 1.73i)35-s + (0.5 + 0.866i)37-s + (−3 − 5.19i)41-s + (−5.5 + 9.52i)43-s + 6·47-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + (0.755 + 0.654i)7-s + (−0.301 − 0.522i)11-s + (0.416 + 0.720i)13-s + (−0.970 + 1.68i)17-s + (0.114 + 0.198i)19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + (−0.371 + 0.643i)29-s + 0.538·31-s + (0.845 − 0.292i)35-s + (0.0821 + 0.142i)37-s + (−0.468 − 0.811i)41-s + (−0.838 + 1.45i)43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.771304709\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.771304709\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 - 1.73i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (4 - 6.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.981943095355619302911644950567, −8.485862272427510297256843424548, −7.907974488086105328692353059732, −6.70603804533987599323116807290, −5.85148063232732749351307855217, −5.33653311906833003067491435070, −4.39495063421745366945109745250, −3.51131856340625239343958472360, −2.02998938511766737556348125152, −1.46153644938947387268722094698,
0.60301918906794572201867742066, 2.17818439064586143987430669843, 2.80951950098308471228437779420, 4.13966417382138711149414154104, 4.81010935044856428897837040456, 5.74896696538666990325079089328, 6.74714563035558892810252706902, 7.19905446495384035248345791112, 8.088055200662331662843651367220, 8.793208803279708033121274967227