L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)13-s + (−0.866 − 0.499i)15-s + (−0.5 − 0.866i)17-s − i·19-s + (0.866 + 0.5i)23-s + i·27-s + (0.5 − 0.866i)29-s + 2i·31-s − 0.999i·39-s + (−0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 − 0.5i)47-s + 49-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)13-s + (−0.866 − 0.499i)15-s + (−0.5 − 0.866i)17-s − i·19-s + (0.866 + 0.5i)23-s + i·27-s + (0.5 − 0.866i)29-s + 2i·31-s − 0.999i·39-s + (−0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 − 0.5i)47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.260453432\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.260453432\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 2iT - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.469381371548169058012118591293, −8.656353934983040406366919718931, −8.380998961346037792458308540163, −7.42198217272792058501705183934, −6.72814245635822106468757900617, −5.30924291287216351748220960892, −4.71071284471956799887229606276, −3.39141442100870189581158695675, −2.57476138032715204162864717431, −1.10050280817860589540909830881,
1.97461444169619673993998852549, 3.21647588991003675215054211196, 3.74725239184903231767558580199, 4.68661615658372845467334466105, 6.18204788887165868167193491315, 6.72552270229868958113104509537, 7.85390698760254343388099242505, 8.462157388716020525785505179617, 9.258738468380851951473213140809, 9.996015271613500717566228374038