| L(s) = 1 | + (−2.64 − 1.41i)3-s + 6.57i·5-s − 2.64·7-s + (5 + 7.48i)9-s − 0.412i·11-s − 20.5·13-s + (9.29 − 17.3i)15-s − 15.8i·17-s − 16·19-s + (7.00 + 3.74i)21-s − 36.0i·23-s − 18.1·25-s + (−2.64 − 26.8i)27-s + 20.8i·29-s + 5.54·31-s + ⋯ |
| L(s) = 1 | + (−0.881 − 0.471i)3-s + 1.31i·5-s − 0.377·7-s + (0.555 + 0.831i)9-s − 0.0374i·11-s − 1.58·13-s + (0.619 − 1.15i)15-s − 0.934i·17-s − 0.842·19-s + (0.333 + 0.178i)21-s − 1.56i·23-s − 0.726·25-s + (−0.0979 − 0.995i)27-s + 0.717i·29-s + 0.178·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8123326479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8123326479\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.64 + 1.41i)T \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 6.57iT - 25T^{2} \) |
| 11 | \( 1 + 0.412iT - 121T^{2} \) |
| 13 | \( 1 + 20.5T + 169T^{2} \) |
| 17 | \( 1 + 15.8iT - 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 + 36.0iT - 529T^{2} \) |
| 29 | \( 1 - 20.8iT - 841T^{2} \) |
| 31 | \( 1 - 5.54T + 961T^{2} \) |
| 37 | \( 1 + 20T + 1.36e3T^{2} \) |
| 41 | \( 1 - 76.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 8.48iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 1.64iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 49.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 87.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 12.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 84.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 4.12iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 31.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 68.8T + 9.40e3T^{2} \) |
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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
−9.720300153695922337136175897246, −8.394497900101611692438182338800, −7.34836862286039149540290381060, −6.86562293153097874960860296376, −6.33055750702095677457550057482, −5.21586903416035968461122256630, −4.39198503541328071346861186700, −2.90139255278240253522825863402, −2.24921459829081839420526539554, −0.41399477248674668551590339949,
0.67736617589139821807116733640, 2.04185945674341608830154622236, 3.73593468534690207832162270594, 4.49621738852940948044476008920, 5.28922411105733328671735052876, 5.90344025076772480895287195920, 6.98332118158880250101925131487, 7.85913858750261612032579694582, 8.945628026207457442408306648943, 9.484809690921081805150730231573
