L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 + 0.342i)3-s + (0.939 − 0.342i)4-s + (−0.984 − 0.173i)6-s + (−0.866 + 0.5i)8-s + (0.766 + 0.642i)9-s + 0.999·12-s + (0.816 − 0.218i)13-s + (0.766 − 0.642i)16-s + (−0.866 − 0.5i)18-s + (0.866 + 0.5i)23-s + (−0.984 + 0.173i)24-s + (0.866 − 0.5i)25-s + (−0.766 + 0.357i)26-s + (0.500 + 0.866i)27-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 + 0.342i)3-s + (0.939 − 0.342i)4-s + (−0.984 − 0.173i)6-s + (−0.866 + 0.5i)8-s + (0.766 + 0.642i)9-s + 0.999·12-s + (0.816 − 0.218i)13-s + (0.766 − 0.642i)16-s + (−0.866 − 0.5i)18-s + (0.866 + 0.5i)23-s + (−0.984 + 0.173i)24-s + (0.866 − 0.5i)25-s + (−0.766 + 0.357i)26-s + (0.500 + 0.866i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.243815853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.243815853\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (-0.816 + 0.218i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (1.10 + 0.296i)T + (0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 + (0.300 + 0.173i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.642 + 1.11i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 1.96iT - T^{2} \) |
| 73 | \( 1 + 0.347iT - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−8.754029856018480025328853323358, −8.419152215327024554303537864597, −7.45694726278428121496289421939, −7.04821833262516299342982678149, −5.98521495749201929633865297117, −5.18967075673598520529205386411, −4.00512015126715851478852903575, −3.16355643913298459942688617204, −2.29226732807475845884701052748, −1.23780399113737841668104884685,
1.13983523022534445265945219637, 2.00326387761382612243509779638, 3.06823180882170467318266117200, 3.61002253465541591317088000406, 4.83406387388398721798920165493, 6.14451095682889919938719576956, 6.75165282498613155091913973266, 7.45274962905432861625630546678, 8.147122594777048762067463198014, 8.760764849519164263478161472725