L(s) = 1 | − 2·5-s + (0.5 − 0.866i)7-s + (−1 − 1.73i)11-s + (2.5 + 2.59i)13-s + (−2 + 3.46i)17-s + (2 − 3.46i)19-s + (−3 − 5.19i)23-s − 25-s + (3 + 5.19i)29-s + 31-s + (−1 + 1.73i)35-s + (−5 − 8.66i)37-s + (−2 − 3.46i)41-s + (0.5 − 0.866i)43-s − 10·47-s + ⋯ |
L(s) = 1 | − 0.894·5-s + (0.188 − 0.327i)7-s + (−0.301 − 0.522i)11-s + (0.693 + 0.720i)13-s + (−0.485 + 0.840i)17-s + (0.458 − 0.794i)19-s + (−0.625 − 1.08i)23-s − 0.200·25-s + (0.557 + 0.964i)29-s + 0.179·31-s + (−0.169 + 0.292i)35-s + (−0.821 − 1.42i)37-s + (−0.312 − 0.541i)41-s + (0.0762 − 0.132i)43-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4443898363\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4443898363\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-2.5 - 2.59i)T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2 + 3.46i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5 - 8.66i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 17T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (8 + 13.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.5 - 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.598234520475740674676217615923, −8.335411928468864051582234880141, −7.30587670157335293185664883591, −6.64141976046326726324264483714, −5.72557613107036189470991479585, −4.58681283914268334123827571653, −3.97641121952049280483950433583, −3.04688622279034951781632327168, −1.67106340441431152553698877524, −0.16494333998265355746661703885,
1.48691449485519588328302864298, 2.85535320718915863571393227804, 3.71046456697862203037740446204, 4.65071493084016238151224472328, 5.48676202140877108409525439728, 6.38996769745844490032447946177, 7.39951864970804093630069469286, 8.017727022876206135052048015258, 8.532426522069057456257371645468, 9.700311327666085871096518759672