L(s) = 1 | − 1.41·2-s − 1.73i·3-s + 2.00·4-s + 2.23i·5-s + 2.44i·6-s − 2.82·8-s − 2.99·9-s − 3.16i·10-s − 1.83·11-s − 3.46i·12-s + 5.40i·13-s + 3.87·15-s + 4.00·16-s − 10.0i·17-s + 4.24·18-s − 9.19i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.447i·5-s + 0.408i·6-s − 0.353·8-s − 0.333·9-s − 0.316i·10-s − 0.167·11-s − 0.288i·12-s + 0.415i·13-s + 0.258·15-s + 0.250·16-s − 0.591i·17-s + 0.235·18-s − 0.483i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2502118847\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2502118847\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.83T + 121T^{2} \) |
| 13 | \( 1 - 5.40iT - 169T^{2} \) |
| 17 | \( 1 + 10.0iT - 289T^{2} \) |
| 19 | \( 1 + 9.19iT - 361T^{2} \) |
| 23 | \( 1 + 0.921T + 529T^{2} \) |
| 29 | \( 1 - 12.5T + 841T^{2} \) |
| 31 | \( 1 - 41.7iT - 961T^{2} \) |
| 37 | \( 1 - 7.28T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8.12T + 1.84e3T^{2} \) |
| 47 | \( 1 - 33.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 104.T + 2.80e3T^{2} \) |
| 59 | \( 1 + 14.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 23.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 14.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 124. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 80.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 154. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 68.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 88.1iT - 9.40e3T^{2} \) |
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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
−9.518193135994199238293699798630, −8.817018683793394638819273122212, −8.015451595033467577460219788705, −7.16653828763953110225243078723, −6.70436261902639947556200660113, −5.75133138132457285124269160989, −4.66972801833989669508385724961, −3.27335820442321022943518991404, −2.41681502359764652284614793362, −1.27780342115790288192527465873,
0.091320257066313708344868576746, 1.48233034040586409781271862750, 2.75412804935969485184542191374, 3.83115721424702111495512865566, 4.81140414176591136239005917569, 5.77999913453516658378531225547, 6.51098451223296215377965104552, 7.81687222557365794027663657125, 8.162095274046152875194051129923, 9.114039347857416582340528261955