L(s) = 1 | + 2.23·5-s − 0.596i·7-s − 9.27i·11-s + 23.5·13-s − 3.97·17-s − 7.04i·19-s + 32.0i·23-s + 5.00·25-s + 35.6·29-s + 59.2i·31-s − 1.33i·35-s + 5.38·37-s − 40.0·41-s + 36.1i·43-s − 74.0i·47-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.0852i·7-s − 0.843i·11-s + 1.80·13-s − 0.233·17-s − 0.370i·19-s + 1.39i·23-s + 0.200·25-s + 1.23·29-s + 1.91i·31-s − 0.0381i·35-s + 0.145·37-s − 0.977·41-s + 0.839i·43-s − 1.57i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.647249029\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.647249029\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
good | 7 | \( 1 + 0.596iT - 49T^{2} \) |
| 11 | \( 1 + 9.27iT - 121T^{2} \) |
| 13 | \( 1 - 23.5T + 169T^{2} \) |
| 17 | \( 1 + 3.97T + 289T^{2} \) |
| 19 | \( 1 + 7.04iT - 361T^{2} \) |
| 23 | \( 1 - 32.0iT - 529T^{2} \) |
| 29 | \( 1 - 35.6T + 841T^{2} \) |
| 31 | \( 1 - 59.2iT - 961T^{2} \) |
| 37 | \( 1 - 5.38T + 1.36e3T^{2} \) |
| 41 | \( 1 + 40.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 36.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 74.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.55T + 2.80e3T^{2} \) |
| 59 | \( 1 + 36.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.73T + 3.72e3T^{2} \) |
| 67 | \( 1 - 69.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 59.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 83.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 65.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 129. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 130.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 93.1T + 9.40e3T^{2} \) |
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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
−8.772011827979898095657112144265, −7.992484310960423160409271973342, −6.92382780069757440459830231070, −6.30469932204510390123160127599, −5.61072431227379995201400962107, −4.79228455178838693880469493650, −3.61264829491098540581364865278, −3.11316126690055487050242996615, −1.71422819424341758564228482540, −0.876763668645578937828161946355,
0.812638282382978495079601897709, 1.89090875468594688123582562423, 2.79884041139181357304821867183, 3.97087825075548268336042827507, 4.56687067054524232804014555917, 5.70513597477297025307149769707, 6.27072641076054555156026933705, 6.94657272987489342410409435447, 8.002102012363035902866837023835, 8.578132320051207285585356596397