| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s − 4·11-s − i·12-s − 2i·13-s + 16-s − i·17-s + i·18-s − 4·19-s + 4i·22-s − 24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s − 0.554i·13-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 0.917·19-s + 0.852i·22-s − 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256666194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256666194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.087910543834248305861046634847, −8.217657119251712969813260519890, −7.80323405699484606229033304585, −6.48525353000009579257296300101, −5.69750401876247945288874825377, −4.72689083643345171935497120211, −4.30393605622137117591546751804, −2.90052049639930777569910855694, −2.63209985225203183245732193592, −0.940197814882960083574972967081,
0.52462297082912301204589856869, 2.05669617851703378174457535933, 3.00107749095560770941295475832, 4.30483768612326852373799885486, 4.96271772592126370217239940426, 5.99743994823797004689189335285, 6.48349120520564199826544232607, 7.35693212743132773923222440700, 8.013495608938935115657463212527, 8.594596012387730146208750059529
