L(s) = 1 | + (0.965 − 0.258i)3-s + (−0.866 − 0.5i)5-s + (0.258 − 0.965i)7-s + (0.866 − 0.499i)9-s + (−0.965 − 0.258i)15-s − i·21-s + (−0.258 + 0.448i)23-s + (0.499 + 0.866i)25-s + (0.707 − 0.707i)27-s − 1.73i·29-s + (−0.707 + 0.707i)35-s + i·41-s − 1.93i·43-s − 45-s + (0.707 − 1.22i)47-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)3-s + (−0.866 − 0.5i)5-s + (0.258 − 0.965i)7-s + (0.866 − 0.499i)9-s + (−0.965 − 0.258i)15-s − i·21-s + (−0.258 + 0.448i)23-s + (0.499 + 0.866i)25-s + (0.707 − 0.707i)27-s − 1.73i·29-s + (−0.707 + 0.707i)35-s + i·41-s − 1.93i·43-s − 45-s + (0.707 − 1.22i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.524283178\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.524283178\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + 1.73iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + 1.93iT - T^{2} \) |
| 47 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.93T + T^{2} \) |
| 89 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.495929422018053494263165958960, −7.84785868452698874761636941674, −7.39593904956077414094597087552, −6.69171930435385134623223553518, −5.52249359596826361000926365054, −4.36989389964526043145010892509, −4.03197076951498769746450011227, −3.16775500966879478819873456069, −1.98227904507205684182959423852, −0.841387865048662697780064150988,
1.67643038866331166870515898474, 2.77333756730893736275259929571, 3.26781626672289329755711746485, 4.30142344564791385132911233912, 4.93629405386982415305082920376, 6.04196219031372488603794921895, 6.93551576954338345845334354747, 7.67092720325917841311286447420, 8.252193520481709175537262691848, 8.896871505967215057978959791974