L(s) = 1 | + 2·2-s + (−4.86 + 1.82i)3-s + 4·4-s + 6.14i·5-s + (−9.72 + 3.65i)6-s − 15.5·7-s + 8·8-s + (20.3 − 17.7i)9-s + 12.2i·10-s − 37.5·11-s + (−19.4 + 7.31i)12-s − 29.6i·13-s − 31.0·14-s + (−11.2 − 29.8i)15-s + 16·16-s − 47.7i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.936 + 0.351i)3-s + 0.5·4-s + 0.549i·5-s + (−0.661 + 0.248i)6-s − 0.837·7-s + 0.353·8-s + (0.752 − 0.658i)9-s + 0.388i·10-s − 1.03·11-s + (−0.468 + 0.175i)12-s − 0.632i·13-s − 0.592·14-s + (−0.193 − 0.514i)15-s + 0.250·16-s − 0.681i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.221630094\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.221630094\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + (4.86 - 1.82i)T \) |
| 59 | \( 1 + (25.0 - 452. i)T \) |
good | 5 | \( 1 - 6.14iT - 125T^{2} \) |
| 7 | \( 1 + 15.5T + 343T^{2} \) |
| 11 | \( 1 + 37.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 47.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 69.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 3.79T + 1.21e4T^{2} \) |
| 29 | \( 1 + 129. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 12.3iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 237. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 336. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 347. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 173.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 99.3iT - 1.48e5T^{2} \) |
| 61 | \( 1 + 764. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 596. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 755. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 360. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 963.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 669.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 337.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 765. iT - 9.12e5T^{2} \) |
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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
−10.80177521876895769673918260582, −10.30874755026457268746739750199, −9.322451429503641971332925518064, −7.59958394690179333575704978578, −6.82720479691414350128720184597, −5.77351020001959942051237822302, −5.09945009456904416697681537434, −3.72155314065960034759689742059, −2.68601452616449243791158252068, −0.39994719825615693355469073497,
1.31084699527669564558581461669, 2.97762023679179977035103221656, 4.49048810394214373634567136742, 5.32121599795442032648396061285, 6.27048465759644224759724680349, 7.11086185400262529211745496847, 8.218421233384132877779788008649, 9.628502057331175912687476066127, 10.50466975052940224340944255999, 11.41969029355867648836888497577