L(s) = 1 | + (−0.866 − 1.5i)3-s + (−1 + 1.73i)9-s − i·11-s + (−1 − 1.73i)17-s + (−0.866 − 0.5i)19-s + (−0.5 + 0.866i)25-s + 1.73·27-s + (−1.5 + 0.866i)33-s + (1.5 − 0.866i)41-s + (−1.73 + i)43-s − 49-s + (−1.73 + 3i)51-s + 1.73i·57-s + (0.866 + 1.5i)59-s + (0.866 − 1.5i)67-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.5i)3-s + (−1 + 1.73i)9-s − i·11-s + (−1 − 1.73i)17-s + (−0.866 − 0.5i)19-s + (−0.5 + 0.866i)25-s + 1.73·27-s + (−1.5 + 0.866i)33-s + (1.5 − 0.866i)41-s + (−1.73 + i)43-s − 49-s + (−1.73 + 3i)51-s + 1.73i·57-s + (0.866 + 1.5i)59-s + (0.866 − 1.5i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5333147518\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5333147518\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
good | 3 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
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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
−9.404338275434048463896808663126, −8.590707903987930337157450131496, −7.68684135028316677525182091881, −6.97711602030787978597865787828, −6.33257708674911452047802337702, −5.53776359687503644312219280328, −4.63205338091576310812121474394, −3.00488165892666332850729556221, −1.92954769839779621893925985129, −0.49780959635767278189143083260,
2.09417791663724743545999534168, 3.77572869709346389763175668568, 4.27618592708093664970233098100, 5.08077694003217113156156636332, 6.10045168952924732326500265181, 6.64648105408904582220075486021, 8.108389503292234681425570323330, 8.823716397578462684019317027308, 9.909127900551651947405969471774, 10.14847671977434335862292516836