L(s) = 1 | + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (2.82 + 2.82i)5-s + (0.707 − 0.707i)6-s + (1.41 − 1.41i)7-s − i·8-s + 1.00i·9-s + (−2.82 + 2.82i)10-s + (0.707 + 0.707i)12-s + 6·13-s + (1.41 + 1.41i)14-s − 4.00i·15-s + 16-s − 1.00·18-s − 4i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (1.26 + 1.26i)5-s + (0.288 − 0.288i)6-s + (0.534 − 0.534i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.894 + 0.894i)10-s + (0.204 + 0.204i)12-s + 1.66·13-s + (0.377 + 0.377i)14-s − 1.03i·15-s + 0.250·16-s − 0.235·18-s − 0.917i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.118779878\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.118779878\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + (-2.82 - 2.82i)T + 5iT^{2} \) |
| 7 | \( 1 + (-1.41 + 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 - 11iT^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.82 - 2.82i)T + 29iT^{2} \) |
| 31 | \( 1 + (4.24 + 4.24i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.82 + 2.82i)T + 37iT^{2} \) |
| 41 | \( 1 + (-7.07 + 7.07i)T - 41iT^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + (-2.82 + 2.82i)T - 61iT^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + (4.24 + 4.24i)T + 71iT^{2} \) |
| 73 | \( 1 + (1.41 + 1.41i)T + 73iT^{2} \) |
| 79 | \( 1 + (-7.07 + 7.07i)T - 79iT^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (4.24 + 4.24i)T + 97iT^{2} \) |
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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
−9.234046011541761291967308079457, −8.686502994507477111413620248978, −7.45864331393173838062733459929, −6.99979779680319481542775363663, −6.20650363944950140985794417729, −5.76276538350345169889763579322, −4.71968002831761964485574728916, −3.51693811816339145109280867160, −2.34655988128672462293583320183, −1.11855930563323365891349631285,
1.18716447005332026001439736456, 1.74716469390857506638793077964, 3.21448737995741067852110916425, 4.32562108031942184072451296094, 5.15822150412737318690895925776, 5.69248826812880945939162036030, 6.39696254576639840421097014132, 8.092885864477195071309509803214, 8.678749113935464235375355975593, 9.284296117231742558428266471757