L(s) = 1 | + 0.554i·2-s + 0.801·3-s + 1.69·4-s + 2.80i·5-s + 0.445i·6-s − 2.69i·7-s + 2.04i·8-s − 2.35·9-s − 1.55·10-s − 1.19i·11-s + 1.35·12-s + 1.49·14-s + 2.24i·15-s + 2.24·16-s − 1.13·17-s − 1.30i·18-s + ⋯ |
L(s) = 1 | + 0.392i·2-s + 0.462·3-s + 0.846·4-s + 1.25i·5-s + 0.181i·6-s − 1.01i·7-s + 0.724i·8-s − 0.785·9-s − 0.491·10-s − 0.361i·11-s + 0.391·12-s + 0.399·14-s + 0.580i·15-s + 0.561·16-s − 0.275·17-s − 0.308i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35538 + 0.578613i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35538 + 0.578613i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 - 0.554iT - 2T^{2} \) |
| 3 | \( 1 - 0.801T + 3T^{2} \) |
| 5 | \( 1 - 2.80iT - 5T^{2} \) |
| 7 | \( 1 + 2.69iT - 7T^{2} \) |
| 11 | \( 1 + 1.19iT - 11T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 + 1.93iT - 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 + 5.89iT - 31T^{2} \) |
| 37 | \( 1 - 0.951iT - 37T^{2} \) |
| 41 | \( 1 + 3.31iT - 41T^{2} \) |
| 43 | \( 1 + 7.15T + 43T^{2} \) |
| 47 | \( 1 + 7.69iT - 47T^{2} \) |
| 53 | \( 1 - 5.87T + 53T^{2} \) |
| 59 | \( 1 + 0.0120iT - 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 - 9.25iT - 67T^{2} \) |
| 71 | \( 1 - 13.7iT - 71T^{2} \) |
| 73 | \( 1 - 12.8iT - 73T^{2} \) |
| 79 | \( 1 - 0.807T + 79T^{2} \) |
| 83 | \( 1 - 16.3iT - 83T^{2} \) |
| 89 | \( 1 + 14.7iT - 89T^{2} \) |
| 97 | \( 1 + 3.13iT - 97T^{2} \) |
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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
−13.23172843026177316489265010637, −11.42699065598586314950194827702, −11.10911071766157646178444943363, −10.10481225846461631473521577550, −8.614244742138536684828859967988, −7.41409192544279426129151621974, −6.86912401101177441015696700849, −5.67399485917868166993388689587, −3.58631967682433352224547082499, −2.50507082103092567003728858686,
1.85363386397674879286724229876, 3.21612061595007245431824799924, 5.02991880792359541495104972319, 6.12411235041285051012484329717, 7.66471625855674818277110366204, 8.778823233459227880407182999975, 9.375482347468606611056271135866, 10.83927107409341649636334428678, 11.86302020321111432584732948672, 12.46411806192441046530082690304