L(s) = 1 | + (−0.979 + 1.02i)2-s + (−1.71 + 0.218i)3-s + (−0.0819 − 1.99i)4-s + (−2.22 − 0.218i)5-s + (1.45 − 1.96i)6-s + 7-s + (2.11 + 1.87i)8-s + (2.90 − 0.750i)9-s + (2.40 − 2.05i)10-s − 2.59·11-s + (0.577 + 3.41i)12-s − 5.21i·13-s + (−0.979 + 1.02i)14-s + (3.87 − 0.110i)15-s + (−3.98 + 0.327i)16-s + 3.91·17-s + ⋯ |
L(s) = 1 | + (−0.692 + 0.721i)2-s + (−0.992 + 0.126i)3-s + (−0.0409 − 0.999i)4-s + (−0.995 − 0.0976i)5-s + (0.595 − 0.803i)6-s + 0.377·7-s + (0.749 + 0.662i)8-s + (0.968 − 0.250i)9-s + (0.759 − 0.650i)10-s − 0.781·11-s + (0.166 + 0.986i)12-s − 1.44i·13-s + (−0.261 + 0.272i)14-s + (0.999 − 0.0286i)15-s + (−0.996 + 0.0819i)16-s + 0.950·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0695 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0695 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.332309 + 0.356289i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332309 + 0.356289i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.979 - 1.02i)T \) |
| 3 | \( 1 + (1.71 - 0.218i)T \) |
| 5 | \( 1 + (2.22 + 0.218i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2.59T + 11T^{2} \) |
| 13 | \( 1 + 5.21iT - 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 5.21iT - 19T^{2} \) |
| 23 | \( 1 - 7.44iT - 23T^{2} \) |
| 29 | \( 1 - 8.44iT - 29T^{2} \) |
| 31 | \( 1 - 1.42iT - 31T^{2} \) |
| 37 | \( 1 - 7.99iT - 37T^{2} \) |
| 41 | \( 1 + 7.49iT - 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 + 4.36iT - 47T^{2} \) |
| 53 | \( 1 + 0.310T + 53T^{2} \) |
| 59 | \( 1 - 2.17T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 - 11.3iT - 73T^{2} \) |
| 79 | \( 1 - 7.09iT - 79T^{2} \) |
| 83 | \( 1 + 0.802iT - 83T^{2} \) |
| 89 | \( 1 - 6.98iT - 89T^{2} \) |
| 97 | \( 1 - 6.80iT - 97T^{2} \) |
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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
−11.18686129924864932658744544605, −10.48479653877268899491746809566, −9.826398109677484537583422631277, −8.341712988030124702625936815325, −7.76416542762094707062573114398, −6.98475491226966629481852199955, −5.42687045717768663619443725524, −5.29770474077448760936383832148, −3.63063418445302744094841976884, −1.09024680608640426700690187883,
0.56782448380520726439078192272, 2.38758534220271275354055975338, 4.08490222112926353840953389430, 4.80596796247445629241940931173, 6.49732892763253092129283073568, 7.42075958428121937000408557103, 8.139066659773861432910890692939, 9.291545151229384260246595330566, 10.31528932434863741403868398048, 11.17061810225289801187377570822