L(s) = 1 | − 2-s + (1.28 − 1.28i)3-s + 4-s + (1.69 − 1.45i)5-s + (−1.28 + 1.28i)6-s + (0.579 − 0.579i)7-s − 8-s − 0.324i·9-s + (−1.69 + 1.45i)10-s − 3.64i·11-s + (1.28 − 1.28i)12-s + 3.10·13-s + (−0.579 + 0.579i)14-s + (0.301 − 4.06i)15-s + 16-s + 8.09i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.744 − 0.744i)3-s + 0.5·4-s + (0.757 − 0.652i)5-s + (−0.526 + 0.526i)6-s + (0.219 − 0.219i)7-s − 0.353·8-s − 0.108i·9-s + (−0.535 + 0.461i)10-s − 1.09i·11-s + (0.372 − 0.372i)12-s + 0.861·13-s + (−0.154 + 0.154i)14-s + (0.0777 − 1.04i)15-s + 0.250·16-s + 1.96i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18565 - 0.798185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18565 - 0.798185i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (-1.69 + 1.45i)T \) |
| 37 | \( 1 + (4.87 + 3.63i)T \) |
good | 3 | \( 1 + (-1.28 + 1.28i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.579 + 0.579i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.64iT - 11T^{2} \) |
| 13 | \( 1 - 3.10T + 13T^{2} \) |
| 17 | \( 1 - 8.09iT - 17T^{2} \) |
| 19 | \( 1 + (3.05 + 3.05i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 + (-1.37 + 1.37i)T - 29iT^{2} \) |
| 31 | \( 1 + (-3.23 - 3.23i)T + 31iT^{2} \) |
| 41 | \( 1 + 9.31iT - 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + (4.11 - 4.11i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.446 - 0.446i)T + 53iT^{2} \) |
| 59 | \( 1 + (6.16 + 6.16i)T + 59iT^{2} \) |
| 61 | \( 1 + (-8.71 - 8.71i)T + 61iT^{2} \) |
| 67 | \( 1 + (2.01 + 2.01i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.00151T + 71T^{2} \) |
| 73 | \( 1 + (9.32 - 9.32i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.760 - 0.760i)T + 79iT^{2} \) |
| 83 | \( 1 + (-3.16 - 3.16i)T + 83iT^{2} \) |
| 89 | \( 1 + (5.80 - 5.80i)T - 89iT^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−10.88535652022666594780323931264, −10.38950949640281381922779247798, −8.965175415954867260264122813232, −8.486093263931210028921060905936, −7.893218569812336127170815288549, −6.46411121868357387533082076327, −5.76587962309845254524752235535, −3.96010636596146139728493430406, −2.32921248830963221834218122604, −1.29472393702820553404810635433,
2.01211419774416913897158231476, 3.10976968900083475850239133301, 4.48044463051748814040576147632, 5.96223810959832223515934673858, 6.94109526641169660255623601579, 8.060932753544573902148772837212, 9.007341150442132330855321480987, 9.867030782584455283647560007711, 10.12872015520957655727483553024, 11.35637349903178541967792769525