| L(s) = 1 | + 5.40·2-s + 13.2·4-s + 16.7i·7-s − 15.1·8-s + 193. i·11-s + 222. i·13-s + 90.3i·14-s − 292.·16-s + 87.8·17-s + 477.·19-s + 1.04e3i·22-s + 627.·23-s + 1.20e3i·26-s + 220. i·28-s − 204. i·29-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.825·4-s + 0.341i·7-s − 0.236·8-s + 1.60i·11-s + 1.31i·13-s + 0.461i·14-s − 1.14·16-s + 0.304·17-s + 1.32·19-s + 2.16i·22-s + 1.18·23-s + 1.77i·26-s + 0.281i·28-s − 0.243i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.292778189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.292778189\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 5.40T + 16T^{2} \) |
| 7 | \( 1 - 16.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 193. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 222. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 87.8T + 8.35e4T^{2} \) |
| 19 | \( 1 - 477.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 627.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 204. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.19e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.38e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.31e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.49e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.94e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.95e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.17e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.29e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.86e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.16e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.09e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 5.20e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 3.13e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.36e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.07e3iT - 8.85e7T^{2} \) |
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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
−12.00747833797615265760767502572, −11.34401716962954382774997935501, −9.774768769914441621774165481897, −9.062461185642455299383812100188, −7.39015979976312575433414713282, −6.55800606147544692364135473681, −5.23167894831791919508396939969, −4.53289609675731914502479376132, −3.27226293483322569574614205622, −1.87968916407029167188440131628,
0.71991359268581069224852677899, 3.01072894160973980434793598352, 3.62762089422574585344763811436, 5.23814600739844766296667936426, 5.71902069128364439921416994449, 7.06339943061908510357409888742, 8.273888579933506189954649819079, 9.431253019851728216886296107355, 10.83664197916847019811968232320, 11.44256754451974194703637154967
