L(s) = 1 | − 3-s + (−0.866 − 0.5i)4-s + 9-s + (−0.366 + 1.36i)11-s + (0.866 + 0.5i)12-s + (0.5 − 0.866i)13-s + (0.499 + 0.866i)16-s − i·17-s + (−1 + i)19-s + (−0.866 − 0.5i)23-s − 27-s + (−0.5 − 0.866i)29-s + (0.366 − 1.36i)33-s + (−0.866 − 0.5i)36-s + (1 + i)37-s + ⋯ |
L(s) = 1 | − 3-s + (−0.866 − 0.5i)4-s + 9-s + (−0.366 + 1.36i)11-s + (0.866 + 0.5i)12-s + (0.5 − 0.866i)13-s + (0.499 + 0.866i)16-s − i·17-s + (−1 + i)19-s + (−0.866 − 0.5i)23-s − 27-s + (−0.5 − 0.866i)29-s + (0.366 − 1.36i)33-s + (−0.866 − 0.5i)36-s + (1 + i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5176053404\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5176053404\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-1 + i)T - iT^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + (-1 - i)T + iT^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T^{2} \) |
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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
−9.444930611255535836425637100917, −8.016628270018677764690130909907, −7.79869212803549420742933351098, −6.42945240434180854140127473613, −6.07801557565012339760402147131, −5.08858830245255673536930640089, −4.58915883096069873916596771733, −3.84899247429965359862985175003, −2.27797494818232771576332496444, −1.04671269292018965834359191463,
0.46716688699010155285830984802, 2.00693177149277525959063363460, 3.59097158342244755556324677901, 4.03329527102991937547660542722, 4.99521539836493012706220854013, 5.75871335249062348630871336349, 6.37860111696882272943935716871, 7.30061274287630028406995192581, 8.168949176125350183546625794781, 8.834843576872041701542852124902