L(s) = 1 | + (0.535 − 0.309i)2-s + (−0.309 − 0.535i)3-s + (−0.309 + 0.535i)4-s + (−0.330 − 0.190i)6-s + (−1.40 − 0.809i)7-s + 0.999i·8-s + (0.309 − 0.535i)9-s + (−0.866 + 0.5i)11-s + 0.381·12-s − 14-s − 0.381i·18-s + (−0.535 − 0.309i)19-s + i·21-s + (−0.309 + 0.535i)22-s + (−0.809 − 1.40i)23-s + (0.535 − 0.309i)24-s + ⋯ |
L(s) = 1 | + (0.535 − 0.309i)2-s + (−0.309 − 0.535i)3-s + (−0.309 + 0.535i)4-s + (−0.330 − 0.190i)6-s + (−1.40 − 0.809i)7-s + 0.999i·8-s + (0.309 − 0.535i)9-s + (−0.866 + 0.5i)11-s + 0.381·12-s − 14-s − 0.381i·18-s + (−0.535 − 0.309i)19-s + i·21-s + (−0.309 + 0.535i)22-s + (−0.809 − 1.40i)23-s + (0.535 − 0.309i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3079718098\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3079718098\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.535 + 0.309i)T + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + (1.40 + 0.809i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.535 + 0.309i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (1.40 - 0.809i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + 0.618iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.61iT - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.078179341100794498149264832471, −8.109289376955329770952741714506, −7.36046988571913609946652446805, −6.62831538189258980785098328560, −5.94965653196301273070783344782, −4.72112316929006317912341035959, −3.98096519049005469572333588841, −3.19745163851123231445037232704, −2.15237883155234490761542186277, −0.18385131770699722780434294840,
2.05986217996456726110407011412, 3.38725381702254621835130983226, 4.08698321424054524812220939389, 5.27746476632407186688189148697, 5.63544413042699576444749361359, 6.30943895161998876408386061435, 7.29092393996187886894299321804, 8.312961777954797807031946997623, 9.261493749808808747137646313007, 9.973000888640401572025764327140