L(s) = 1 | + (−0.366 + 0.366i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.866 − 0.5i)17-s + 0.732i·25-s + (1.5 + 0.866i)29-s + (1.86 − 0.5i)37-s + (1.86 − 0.5i)41-s + (0.5 + 0.133i)45-s + (−0.866 − 0.5i)49-s + i·53-s + (−0.866 + 0.5i)61-s + (−0.133 − 0.5i)65-s + (1.36 − 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯ |
L(s) = 1 | + (−0.366 + 0.366i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.866 − 0.5i)17-s + 0.732i·25-s + (1.5 + 0.866i)29-s + (1.86 − 0.5i)37-s + (1.86 − 0.5i)41-s + (0.5 + 0.133i)45-s + (−0.866 − 0.5i)49-s + i·53-s + (−0.866 + 0.5i)61-s + (−0.133 − 0.5i)65-s + (1.36 − 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.133156686\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.133156686\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - iT - T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.104241063794921509645928883540, −7.954222561041058188154798089519, −7.40084114066933685431148503648, −6.60064729606784527216270860115, −5.95590815885459182680499672331, −4.99080894977485011405071949089, −4.13542678883333485709505908580, −3.25578719646000899301107582398, −2.52429185860616203246947830923, −1.02724387459941760741150787824,
0.892198906343324535952034737952, 2.38815573006241071183332501094, 3.07440001321462754717056425364, 4.29823811408053858463768280750, 4.86635912528479448975027052556, 5.77454395282607514960973054792, 6.37851587140962471785008619214, 7.66656119230051686908066232746, 7.978129928264267928041697858084, 8.496035716560202743467224032704