L(s) = 1 | + (0.991 − 0.130i)5-s + (−0.866 + 0.5i)9-s + (−1.30 + 1.30i)13-s + (0.739 − 0.198i)17-s + (0.965 − 0.258i)25-s + 1.41i·29-s + (1.93 + 0.517i)37-s + 1.84i·41-s + (−0.793 + 0.608i)45-s + (1.36 − 0.366i)53-s + (−0.662 + 0.382i)61-s + (−1.12 + 1.46i)65-s + (−0.198 − 0.739i)73-s + (0.499 − 0.866i)81-s + (0.707 − 0.292i)85-s + ⋯ |
L(s) = 1 | + (0.991 − 0.130i)5-s + (−0.866 + 0.5i)9-s + (−1.30 + 1.30i)13-s + (0.739 − 0.198i)17-s + (0.965 − 0.258i)25-s + 1.41i·29-s + (1.93 + 0.517i)37-s + 1.84i·41-s + (−0.793 + 0.608i)45-s + (1.36 − 0.366i)53-s + (−0.662 + 0.382i)61-s + (−1.12 + 1.46i)65-s + (−0.198 − 0.739i)73-s + (0.499 − 0.866i)81-s + (0.707 − 0.292i)85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.329987573\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.329987573\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.991 + 0.130i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 17 | \( 1 + (-0.739 + 0.198i)T + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - 1.84iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
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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
−8.869468433389075205640564751079, −8.057956967225917975266430304013, −7.24974522090137060471140184941, −6.51336881562603744127750521525, −5.75526208065868474551141263559, −5.02855750825867026041379629763, −4.45168176111498081840633512469, −3.02842921073400671738883633525, −2.42049997351164693120069882949, −1.41574069744748303335841413108,
0.76041929258738686985282340500, 2.33805202336877140262489564712, 2.78818017277855577153989438057, 3.84254203410227553659008452704, 4.99438777957034746616791217312, 5.74340856966119873623902851084, 6.00908453497815629050060946660, 7.13131491483062938317060670129, 7.78642008893395940878546621387, 8.543017547024085620746081597387