L(s) = 1 | + (18.1 + 61.3i)2-s − 420. i·3-s + (−3.43e3 + 2.23e3i)4-s − 204.·5-s + (2.58e4 − 7.65e3i)6-s − 1.21e5i·7-s + (−1.99e5 − 1.70e5i)8-s − 1.77e5·9-s + (−3.71e3 − 1.25e4i)10-s − 2.28e6i·11-s + (9.38e5 + 1.44e6i)12-s + 6.03e6·13-s + (7.46e6 − 2.21e6i)14-s + 8.60e4i·15-s + (6.82e6 − 1.53e7i)16-s − 1.11e7·17-s + ⋯ |
L(s) = 1 | + (0.284 + 0.958i)2-s − 0.577i·3-s + (−0.838 + 0.544i)4-s − 0.0130·5-s + (0.553 − 0.163i)6-s − 1.03i·7-s + (−0.760 − 0.649i)8-s − 0.333·9-s + (−0.00371 − 0.0125i)10-s − 1.29i·11-s + (0.314 + 0.484i)12-s + 1.24·13-s + (0.991 − 0.293i)14-s + 0.00755i·15-s + (0.406 − 0.913i)16-s − 0.459·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(1.21081 - 0.657396i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21081 - 0.657396i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-18.1 - 61.3i)T \) |
| 3 | \( 1 + 420. iT \) |
good | 5 | \( 1 + 204.T + 2.44e8T^{2} \) |
| 7 | \( 1 + 1.21e5iT - 1.38e10T^{2} \) |
| 11 | \( 1 + 2.28e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 - 6.03e6T + 2.32e13T^{2} \) |
| 17 | \( 1 + 1.11e7T + 5.82e14T^{2} \) |
| 19 | \( 1 + 5.58e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 - 7.22e7iT - 2.19e16T^{2} \) |
| 29 | \( 1 + 7.70e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + 3.21e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 + 3.35e9T + 6.58e18T^{2} \) |
| 41 | \( 1 - 6.76e9T + 2.25e19T^{2} \) |
| 43 | \( 1 + 3.68e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 - 7.64e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 - 2.39e10T + 4.91e20T^{2} \) |
| 59 | \( 1 - 6.16e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 8.37e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + 1.69e11iT - 8.18e21T^{2} \) |
| 71 | \( 1 + 1.22e11iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 1.26e11T + 2.29e22T^{2} \) |
| 79 | \( 1 - 3.45e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 - 2.27e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 7.45e11T + 2.46e23T^{2} \) |
| 97 | \( 1 - 1.04e12T + 6.93e23T^{2} \) |
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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
−16.88234484218738008166638202786, −15.66509148055617887596867498671, −13.82379509027578550690990012036, −13.30858750426571472950151357568, −11.18247524506046164157379058973, −8.805663290932218951853982449279, −7.35581997111206688881884029605, −5.92081732381062836489849017978, −3.74432113706509519101254835246, −0.58570228701622891830256652540,
2.00796309419130781804506261026, 3.92894857835395170367161808835, 5.64565545645210864581507177346, 8.742461304270461771127594967501, 10.06007585837038815678218377906, 11.55881506292851018293414947061, 12.82205971428195458504249198987, 14.54167574337400936299189490887, 15.70523418278996592095396741235, 17.75416881194216512537173929109