L(s) = 1 | − 1.41·2-s + (0.396 − 2.97i)3-s + 2.00·4-s + (−0.560 + 4.20i)6-s + 2.64i·7-s − 2.82·8-s + (−8.68 − 2.35i)9-s + 2.01i·11-s + (0.792 − 5.94i)12-s + 11.6i·13-s − 3.74i·14-s + 4.00·16-s + 17.3·17-s + (12.2 + 3.33i)18-s − 36.1·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.132 − 0.991i)3-s + 0.500·4-s + (−0.0934 + 0.700i)6-s + 0.377i·7-s − 0.353·8-s + (−0.965 − 0.261i)9-s + 0.183i·11-s + (0.0660 − 0.495i)12-s + 0.892i·13-s − 0.267i·14-s + 0.250·16-s + 1.02·17-s + (0.682 + 0.185i)18-s − 1.90·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.253529806\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.253529806\) |
\(L(2)\) |
|
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 | \( 1 + (-0.396 + 2.97i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 11 | \( 1 - 2.01iT - 121T^{2} \) |
| 13 | \( 1 - 11.6iT - 169T^{2} \) |
| 17 | \( 1 - 17.3T + 289T^{2} \) |
| 19 | \( 1 + 36.1T + 361T^{2} \) |
| 23 | \( 1 - 32.2T + 529T^{2} \) |
| 29 | \( 1 + 46.1iT - 841T^{2} \) |
| 31 | \( 1 - 34.0T + 961T^{2} \) |
| 37 | \( 1 + 31.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 92.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 18.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 45.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 33.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 25.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 48.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 32.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 82.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 48.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 14.5iT - 9.40e3T^{2} \) |
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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
−9.319385029565195371805329195781, −8.671262838873875735126682793881, −7.984773714675155983904923969320, −7.08510022423622801129182502275, −6.42672126972114195786952759423, −5.58796835631516047912619695232, −4.16263248108740637485628805346, −2.71183726000413153488563830781, −1.93224824938949396608742759134, −0.63646462683774696571236643329,
0.911288649124523111326356442464, 2.64563310871452445210797983845, 3.49708092070486335739781557371, 4.64153128984747792697537967870, 5.56997840619008898044740587405, 6.55621858964542782898981901627, 7.61950076610930832688065436293, 8.478066308213553925282449660395, 8.986005808973802882337087292149, 10.00405541910743619604749108140