L(s) = 1 | + (0.540 − 0.540i)2-s + (0.0858 − 1.72i)3-s + 1.41i·4-s + (−0.996 + 0.996i)5-s + (−0.888 − 0.981i)6-s + (−1.80 + 1.80i)7-s + (1.84 + 1.84i)8-s + (−2.98 − 0.296i)9-s + 1.07i·10-s + (3.35 + 3.35i)11-s + (2.44 + 0.121i)12-s + 1.94i·14-s + (1.63 + 1.80i)15-s − 0.837·16-s + 5.80·17-s + (−1.77 + 1.45i)18-s + ⋯ |
L(s) = 1 | + (0.382 − 0.382i)2-s + (0.0495 − 0.998i)3-s + 0.708i·4-s + (−0.445 + 0.445i)5-s + (−0.362 − 0.400i)6-s + (−0.681 + 0.681i)7-s + (0.652 + 0.652i)8-s + (−0.995 − 0.0989i)9-s + 0.340i·10-s + (1.01 + 1.01i)11-s + (0.707 + 0.0350i)12-s + 0.520i·14-s + (0.422 + 0.467i)15-s − 0.209·16-s + 1.40·17-s + (−0.417 + 0.342i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42310 + 0.433156i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42310 + 0.433156i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0858 + 1.72i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.540 + 0.540i)T - 2iT^{2} \) |
| 5 | \( 1 + (0.996 - 0.996i)T - 5iT^{2} \) |
| 7 | \( 1 + (1.80 - 1.80i)T - 7iT^{2} \) |
| 11 | \( 1 + (-3.35 - 3.35i)T + 11iT^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 + (-2.39 - 2.39i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.39T + 23T^{2} \) |
| 29 | \( 1 + 6.57iT - 29T^{2} \) |
| 31 | \( 1 + (-0.386 - 0.386i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.93 - 5.93i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.734 + 0.734i)T - 41iT^{2} \) |
| 43 | \( 1 + 7.56iT - 43T^{2} \) |
| 47 | \( 1 + (0.243 + 0.243i)T + 47iT^{2} \) |
| 53 | \( 1 + 2.07iT - 53T^{2} \) |
| 59 | \( 1 + (-3.56 - 3.56i)T + 59iT^{2} \) |
| 61 | \( 1 - 7.04T + 61T^{2} \) |
| 67 | \( 1 + (4.54 + 4.54i)T + 67iT^{2} \) |
| 71 | \( 1 + (-6.79 + 6.79i)T - 71iT^{2} \) |
| 73 | \( 1 + (6.04 - 6.04i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.77T + 79T^{2} \) |
| 83 | \( 1 + (-8.31 + 8.31i)T - 83iT^{2} \) |
| 89 | \( 1 + (9.62 + 9.62i)T + 89iT^{2} \) |
| 97 | \( 1 + (1.34 + 1.34i)T + 97iT^{2} \) |
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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
−11.57204169730123188399084273784, −10.14707845289313498152139444710, −9.148390504347017221042642492759, −8.056441747850551057763148620383, −7.40530431414829387658085764475, −6.55998981957764279685297783178, −5.44607742951961216598902900282, −3.84985085079946345045218983038, −3.06656302415721435796500403872, −1.84504432928248675472987329106,
0.829453400482372617546343041579, 3.38771292485780837696330724464, 4.06830799062652429504480821036, 5.17087890286083247599209811552, 5.98788137741352504133381921327, 6.97556553950129406620480090667, 8.237703727052165537625115458260, 9.250985379481922067796850113229, 9.914868293739665126856769720498, 10.69737415434854382419115955656