L(s) = 1 | + (−0.707 + 0.707i)2-s − i·3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)8-s − 9-s − 1.00·10-s − 1.00·12-s + (0.707 − 0.707i)13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (1 − i)17-s + (0.707 − 0.707i)18-s + (0.707 − 0.707i)20-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − i·3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)8-s − 9-s − 1.00·10-s − 1.00·12-s + (0.707 − 0.707i)13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (1 − i)17-s + (0.707 − 0.707i)18-s + (0.707 − 0.707i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8924742559\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8924742559\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.470729418092282604234724843851, −8.709574448191944530536118940961, −7.77172419402153112101484163210, −7.31346005294470164627599281141, −6.45876255872933951486078947224, −5.81336747421711330184063362126, −5.18128536952878359356792499441, −3.29981189407777004092295128691, −2.25147688489746114484075017064, −1.07937541647833541594662425512,
1.34772033974007551188274029329, 2.53586982846714535759586505236, 3.76595838011834128611490361584, 4.32909344991039553932524331509, 5.52238146006170480284387723841, 6.23667599481170632077665688881, 7.65143691252782783389436353054, 8.413344674094134084033478539952, 9.223174896378582187110482700975, 9.465378944884983541507692861006