L(s) = 1 | + 1.73·3-s + 2i·4-s + (3.09 − 3.09i)7-s + 2.99·9-s + 3.46i·12-s − 4·16-s + (−2.26 − 2.26i)19-s + (5.36 − 5.36i)21-s + 5i·25-s + 5.19·27-s + (6.19 + 6.19i)28-s + (0.830 + 0.830i)31-s + 5.99i·36-s + (−8.46 + 8.46i)37-s − 1.73i·43-s + ⋯ |
L(s) = 1 | + 1.00·3-s + i·4-s + (1.17 − 1.17i)7-s + 0.999·9-s + 0.999i·12-s − 16-s + (−0.520 − 0.520i)19-s + (1.17 − 1.17i)21-s + i·25-s + 1.00·27-s + (1.17 + 1.17i)28-s + (0.149 + 0.149i)31-s + 0.999i·36-s + (−1.39 + 1.39i)37-s − 0.264i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.17242 + 0.321668i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17242 + 0.321668i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2iT^{2} \) |
| 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 + (-3.09 + 3.09i)T - 7iT^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (2.26 + 2.26i)T + 19iT^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-0.830 - 0.830i)T + 31iT^{2} \) |
| 37 | \( 1 + (8.46 - 8.46i)T - 37iT^{2} \) |
| 41 | \( 1 - 41iT^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + (11.5 + 11.5i)T + 67iT^{2} \) |
| 71 | \( 1 - 71iT^{2} \) |
| 73 | \( 1 + (7.63 - 7.63i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 89iT^{2} \) |
| 97 | \( 1 + (7.02 + 7.02i)T + 97iT^{2} \) |
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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
−10.95296835000448805075020943361, −10.05911361376090806514910565905, −8.845174159931809766541751142262, −8.278336986794601750992838545987, −7.43921663760680981755768897117, −6.89921467456461448234756600202, −4.85577079397094602955820392245, −4.08615422428834798132064129461, −3.10626912948006536621911433562, −1.67722203884899974366815108604,
1.65867033055484333747565087492, 2.49641399468138284479520037360, 4.20998871526297960127435630745, 5.19014445140734830202037017848, 6.13600869747433017216839382978, 7.38318813135460189955454378731, 8.493162812494041066391399964481, 8.882016183464391243933687964247, 9.949215518502308110413263137817, 10.70231519726415158838410462051