L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.23i·5-s − 5.52i·7-s + 2.82·8-s − 3.16i·10-s + 3.99i·11-s + 8.18·13-s − 7.81i·14-s + 4.00·16-s − 17.3i·17-s − 11.0i·19-s − 4.47i·20-s + 5.65i·22-s + (22.7 + 3.11i)23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.447i·5-s − 0.789i·7-s + 0.353·8-s − 0.316i·10-s + 0.363i·11-s + 0.629·13-s − 0.558i·14-s + 0.250·16-s − 1.02i·17-s − 0.583i·19-s − 0.223i·20-s + 0.257i·22-s + (0.990 + 0.135i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.971392124\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.971392124\) |
\(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-22.7 - 3.11i)T \) |
good | 7 | \( 1 + 5.52iT - 49T^{2} \) |
| 11 | \( 1 - 3.99iT - 121T^{2} \) |
| 13 | \( 1 - 8.18T + 169T^{2} \) |
| 17 | \( 1 + 17.3iT - 289T^{2} \) |
| 19 | \( 1 + 11.0iT - 361T^{2} \) |
| 29 | \( 1 + 10.3T + 841T^{2} \) |
| 31 | \( 1 + 20.6T + 961T^{2} \) |
| 37 | \( 1 + 39.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 73.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 52.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 74.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 18.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 6.77T + 3.48e3T^{2} \) |
| 61 | \( 1 + 115. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 30.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 89.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 141. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 66.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 54.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.89iT - 9.40e3T^{2} \) |
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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
−8.840008710792648484285411540540, −7.65924538251858276018435420352, −7.18219015008458469431845166463, −6.35218325885075253832190680825, −5.31960745820425877807984952390, −4.71476083279840742190152414817, −3.85650595983836897000807436131, −2.98592996184483647636048979631, −1.72672897665837577049850722533, −0.57236917762677251399207970469,
1.40278641656560751161461823495, 2.48083761704629781512149017893, 3.40764052886650996144692387221, 4.11655256026805176218362310788, 5.38361641631914922177577905533, 5.81803817461270044154449288354, 6.66062670593093361326884399794, 7.42031862418278148829463276814, 8.534012349277663903362804678113, 8.865731758581501000964680635765