L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s − i·19-s + (1.36 + 0.366i)23-s + (−0.499 + 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.5 − 0.866i)31-s + (1 + i)37-s + (0.5 + 0.866i)41-s + (0.366 + 1.36i)43-s + (−1.36 + 0.366i)47-s + (−0.866 + 0.5i)49-s − 0.999·55-s + (−0.866 + 0.5i)59-s + (0.366 − 1.36i)67-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s − i·19-s + (1.36 + 0.366i)23-s + (−0.499 + 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.5 − 0.866i)31-s + (1 + i)37-s + (0.5 + 0.866i)41-s + (0.366 + 1.36i)43-s + (−1.36 + 0.366i)47-s + (−0.866 + 0.5i)49-s − 0.999·55-s + (−0.866 + 0.5i)59-s + (0.366 − 1.36i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.313174825\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.313174825\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
good | 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + iT - T^{2} \) |
| 97 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
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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
−9.255008774498635435901486109127, −8.044737367527346510793277250004, −7.46342489130258909465145264299, −6.68149533237909890704442245731, −6.15594655693129422097387372010, −5.03727250548567145479746217111, −4.52928294269600174135491041301, −3.11225501615186262238180092418, −2.67460998229789301498853002286, −1.47030529950307616056193297520,
0.840417013288677065313478934405, 2.02660684366316742562794889993, 3.07964816127355457462238995274, 4.05294430640696759182948388416, 5.02664679204751948102298951012, 5.57489389887609852049067239705, 6.31134578130227363861355764636, 7.23653700364907522936559028184, 8.193501684757790226374484954349, 8.617094096108039975615355071409