| L(s) = 1 | + 1.88i·5-s − 10.9·7-s − 0.171i·11-s + 6.48·13-s − 27.2i·17-s − 5.32·19-s + 3.51i·23-s + 21.4·25-s + 34.8i·29-s + 26.9·31-s − 20.6i·35-s + 46.4·37-s + 23.5i·41-s + 55.1·43-s + 57.5i·47-s + ⋯ |
| L(s) = 1 | + 0.376i·5-s − 1.56·7-s − 0.0155i·11-s + 0.498·13-s − 1.60i·17-s − 0.280·19-s + 0.152i·23-s + 0.858·25-s + 1.20i·29-s + 0.869·31-s − 0.590i·35-s + 1.25·37-s + 0.573i·41-s + 1.28·43-s + 1.22i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.442062614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442062614\) |
| \(L(2)\) |
|
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 | \( 1 - 1.88iT - 25T^{2} \) |
| 7 | \( 1 + 10.9T + 49T^{2} \) |
| 11 | \( 1 + 0.171iT - 121T^{2} \) |
| 13 | \( 1 - 6.48T + 169T^{2} \) |
| 17 | \( 1 + 27.2iT - 289T^{2} \) |
| 19 | \( 1 + 5.32T + 361T^{2} \) |
| 23 | \( 1 - 3.51iT - 529T^{2} \) |
| 29 | \( 1 - 34.8iT - 841T^{2} \) |
| 31 | \( 1 - 26.9T + 961T^{2} \) |
| 37 | \( 1 - 46.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 23.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 57.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 51.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 82.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 79.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 44.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 41.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 66.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 115.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 36.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 158. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 62.8T + 9.40e3T^{2} \) |
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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
−9.760411849708566774431637346980, −9.386590002914787444837036452428, −8.345792343933395192636052428559, −7.13623238355346575473616943769, −6.63733799672119416917329958584, −5.74340321427574254263740105806, −4.54643681612988669764089764156, −3.29708197305782856771017528863, −2.68856946886626941928521226355, −0.73480391396487983073673985239,
0.78861011338591634686672295643, 2.44391886440401997528879381507, 3.60249226908947890093818900983, 4.40863312275037803803339089416, 5.95494223753166354820812329959, 6.24380773419524322986053771888, 7.36636017514808643528633264908, 8.467344448494012292003186713318, 9.066264165416839989525358097738, 10.05808583350223028705879074678
