| L(s) = 1 | − 2.16i·5-s + 6.23·7-s − 14.9i·11-s + 13.0·13-s + 3.88i·17-s − 12.7·19-s − 35.9i·23-s + 20.3·25-s + 11.0i·29-s + 36.0·31-s − 13.5i·35-s + 32.6·37-s + 22.5i·41-s − 61.9·43-s + 3.41i·47-s + ⋯ |
| L(s) = 1 | − 0.433i·5-s + 0.890·7-s − 1.35i·11-s + 1.00·13-s + 0.228i·17-s − 0.673·19-s − 1.56i·23-s + 0.812·25-s + 0.382i·29-s + 1.16·31-s − 0.385i·35-s + 0.883·37-s + 0.550i·41-s − 1.44·43-s + 0.0726i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.357604059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.357604059\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.16iT - 25T^{2} \) |
| 7 | \( 1 - 6.23T + 49T^{2} \) |
| 11 | \( 1 + 14.9iT - 121T^{2} \) |
| 13 | \( 1 - 13.0T + 169T^{2} \) |
| 17 | \( 1 - 3.88iT - 289T^{2} \) |
| 19 | \( 1 + 12.7T + 361T^{2} \) |
| 23 | \( 1 + 35.9iT - 529T^{2} \) |
| 29 | \( 1 - 11.0iT - 841T^{2} \) |
| 31 | \( 1 - 36.0T + 961T^{2} \) |
| 37 | \( 1 - 32.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 22.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 61.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 3.41iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 79.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 21.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 78.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 110.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 82.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 6.19T + 5.32e3T^{2} \) |
| 79 | \( 1 - 64.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 112. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 55.4T + 9.40e3T^{2} \) |
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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
−8.385526724928584550491250297921, −8.071561829584510010156499169513, −6.61857761514868893649022013208, −6.25338670630024217260285238117, −5.20147598301040213447521048313, −4.56955991394571327384120489077, −3.64522671477167936207651318320, −2.66194142141981616951227162421, −1.43825668669841260486824451400, −0.58194861663126745242931031585,
1.21134952424500988941895158058, 2.04858887226578748701272677090, 3.10425114806208564558906539685, 4.18650752664054152899262853073, 4.77915868195249157001237186897, 5.72200578634347873361086895924, 6.58360717298798946441710158753, 7.30347165492687753654514299692, 7.970804129477540924377855306120, 8.707364000996916096519502871215
