L(s) = 1 | + (−1 − i)2-s + 3-s + 2i·4-s + (−2 + i)5-s + (−1 − i)6-s + i·7-s + (2 − 2i)8-s + 9-s + (3 + i)10-s + 2i·12-s − 6·13-s + (1 − i)14-s + (−2 + i)15-s − 4·16-s − 2i·17-s + (−1 − i)18-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + 0.577·3-s + i·4-s + (−0.894 + 0.447i)5-s + (−0.408 − 0.408i)6-s + 0.377i·7-s + (0.707 − 0.707i)8-s + 0.333·9-s + (0.948 + 0.316i)10-s + 0.577i·12-s − 1.66·13-s + (0.267 − 0.267i)14-s + (−0.516 + 0.258i)15-s − 16-s − 0.485i·17-s + (−0.235 − 0.235i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.827587621863316540391035609981, −8.926696698067073916940521242340, −8.076355883728250884671638837610, −7.49802308981296106456263057498, −6.70366099803928149041269369318, −4.95540436795898345333706998985, −3.91565065162201978974311969057, −2.96577385877494264927894341267, −2.07327500427967950594078941496, 0,
1.71665184902814543981883396650, 3.31940275563986170186367689426, 4.62861318454356888034716074040, 5.26565231921234808176732281286, 6.85490533945410981959665742012, 7.35913607684984123440333953142, 8.045482269095616553945119907987, 8.938844465520380847061105951189, 9.524576000525702766417180619032