L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + 0.999i·12-s − i·13-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)19-s − 0.999·21-s + 0.999i·27-s + (−0.866 − 0.499i)28-s + 2·31-s + (−0.499 + 0.866i)36-s + (−1.73 − i)37-s + (0.5 − 0.866i)39-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + 0.999i·12-s − i·13-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)19-s − 0.999·21-s + 0.999i·27-s + (−0.866 − 0.499i)28-s + 2·31-s + (−0.499 + 0.866i)36-s + (−1.73 − i)37-s + (0.5 − 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.358490199\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.358490199\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - 2T + T^{2} \) |
| 37 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
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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
−10.34310638821380455449856953471, −9.409615506310209289163852114773, −8.670703045732724951133760637833, −8.029569941269200585931172766697, −7.13190662600404668670397688462, −6.26367363529449067205461166082, −5.00450107030999870098180458543, −3.84391005166167262508875013627, −3.02096976755714187910981716854, −2.34771797430569818960100333045,
1.34768790193108976191285941272, 2.47197013082709774899791691400, 3.57494138434202337835865687000, 4.66490929315396902704700856610, 6.18999004559644605345422925579, 6.58222945416412569508980364314, 7.38687004474828700368709590707, 8.415800432743590563584239387009, 9.316273731660792870832350777994, 10.00071826969149236424756583212