| L(s) = 1 | + (−2.87 − 0.864i)3-s − 2.23i·5-s − 9.02·7-s + (7.50 + 4.96i)9-s − 21.8i·11-s − 21.6·13-s + (−1.93 + 6.42i)15-s + 12.1i·17-s + 3.03·19-s + (25.9 + 7.80i)21-s − 28.5i·23-s − 5.00·25-s + (−17.2 − 20.7i)27-s + 12.0i·29-s − 2.19·31-s + ⋯ |
| L(s) = 1 | + (−0.957 − 0.288i)3-s − 0.447i·5-s − 1.28·7-s + (0.833 + 0.551i)9-s − 1.98i·11-s − 1.66·13-s + (−0.128 + 0.428i)15-s + 0.712i·17-s + 0.159·19-s + (1.23 + 0.371i)21-s − 1.24i·23-s − 0.200·25-s + (−0.639 − 0.768i)27-s + 0.415i·29-s − 0.0706·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2009003095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2009003095\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.87 + 0.864i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 + 9.02T + 49T^{2} \) |
| 11 | \( 1 + 21.8iT - 121T^{2} \) |
| 13 | \( 1 + 21.6T + 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 - 3.03T + 361T^{2} \) |
| 23 | \( 1 + 28.5iT - 529T^{2} \) |
| 29 | \( 1 - 12.0iT - 841T^{2} \) |
| 31 | \( 1 + 2.19T + 961T^{2} \) |
| 37 | \( 1 + 0.839T + 1.36e3T^{2} \) |
| 41 | \( 1 + 35.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 12.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 22.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 9.13iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 80.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 57.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 63.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 52.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 7.46T + 6.24e3T^{2} \) |
| 83 | \( 1 - 82.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 27.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 114.T + 9.40e3T^{2} \) |
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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
−10.17017921311092467699939400706, −9.222901098904233833419105623030, −8.355313549622027604905638747428, −7.30064312054424419482096268747, −6.41705031929664853298037007495, −5.81724869328884684610637334347, −4.93524557734916618404855529878, −3.73182672279438773294442913933, −2.55996656311861038894842934195, −0.78587292439252100397156662139,
0.10227440640908283259114074429, 2.07082349198779070335150999462, 3.30713359534924550787519258253, 4.52569395372309572514317351319, 5.19822333924117204637947351224, 6.33737140153061263082386098689, 7.15497294862908338636173232283, 7.43914205441669889569929853135, 9.427970222689681971034946415487, 9.832336325855908077903667314623
