L(s) = 1 | + 1.73i·3-s − 5.37i·5-s − 2.99·9-s − 8.59·11-s + 21.0i·13-s + 9.31·15-s + 5.48i·17-s + 7.24i·19-s + 28.0·23-s − 3.94·25-s − 5.19i·27-s + 40.3·29-s + 40.5i·31-s − 14.8i·33-s + 66.6·37-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.07i·5-s − 0.333·9-s − 0.781·11-s + 1.61i·13-s + 0.621·15-s + 0.322i·17-s + 0.381i·19-s + 1.21·23-s − 0.157·25-s − 0.192i·27-s + 1.39·29-s + 1.30i·31-s − 0.450i·33-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.512869610\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.512869610\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 5.37iT - 25T^{2} \) |
| 11 | \( 1 + 8.59T + 121T^{2} \) |
| 13 | \( 1 - 21.0iT - 169T^{2} \) |
| 17 | \( 1 - 5.48iT - 289T^{2} \) |
| 19 | \( 1 - 7.24iT - 361T^{2} \) |
| 23 | \( 1 - 28.0T + 529T^{2} \) |
| 29 | \( 1 - 40.3T + 841T^{2} \) |
| 31 | \( 1 - 40.5iT - 961T^{2} \) |
| 37 | \( 1 - 66.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 0.932T + 1.84e3T^{2} \) |
| 47 | \( 1 - 85.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 44.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 63.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 32.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 47.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 14.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 140. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 122.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 33.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 36.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.2iT - 9.40e3T^{2} \) |
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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
−10.63230495318925358783170947646, −9.643465959650558812878768499046, −8.926091362794485802751711523596, −8.291465650574302823090911552099, −7.07720340918578777345128757168, −5.94461858689133057822367696766, −4.81951469666562413666924881204, −4.35310990858762260953602851673, −2.84667863718987127944023087408, −1.25967204220419888401513835563,
0.64013305115799304262993404935, 2.60170764961301906191198090889, 3.11472471768905324108377014330, 4.85427104560463411597489384600, 5.89285626785841312477036838337, 6.79205998293934506310967073838, 7.64831257020186594739170197219, 8.275275368137520670676619220688, 9.596441603681292004991477230453, 10.51396580494932769444838644652