L(s) = 1 | + 1.73i·3-s − 7.41·5-s − 2.99·9-s − 20.1i·11-s − 21.8·13-s − 12.8i·15-s − 13.0·17-s + 14.6i·19-s − 40.9i·23-s + 29.9·25-s − 5.19i·27-s + 32.4·29-s − 51.0i·31-s + 34.9·33-s − 33.9·37-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.48·5-s − 0.333·9-s − 1.83i·11-s − 1.68·13-s − 0.856i·15-s − 0.768·17-s + 0.773i·19-s − 1.78i·23-s + 1.19·25-s − 0.192i·27-s + 1.12·29-s − 1.64i·31-s + 1.05·33-s − 0.917·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3315991834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3315991834\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 7.41T + 25T^{2} \) |
| 11 | \( 1 + 20.1iT - 121T^{2} \) |
| 13 | \( 1 + 21.8T + 169T^{2} \) |
| 17 | \( 1 + 13.0T + 289T^{2} \) |
| 19 | \( 1 - 14.6iT - 361T^{2} \) |
| 23 | \( 1 + 40.9iT - 529T^{2} \) |
| 29 | \( 1 - 32.4T + 841T^{2} \) |
| 31 | \( 1 + 51.0iT - 961T^{2} \) |
| 37 | \( 1 + 33.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 8.10T + 1.68e3T^{2} \) |
| 43 | \( 1 - 69.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 28.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 72.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 4.40iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 33.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 30.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 27.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 61.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 90.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 5.24iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 56.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 123.T + 9.40e3T^{2} \) |
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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
−8.809577564289893528929718191885, −8.216163490309694068374825189690, −7.72389324654930794781817932799, −6.66404687424718749624420335423, −5.88655428715285030252257031223, −4.69388079962380700039544758251, −4.30465017845430894763184470313, −3.28282983091396474947910600143, −2.58829430661141735741261377578, −0.57374048618133055322556145758,
0.14286424678677579197503860576, 1.70931202714474078684361452503, 2.68205966976484731410161240166, 3.74580631062668687812234778852, 4.77061235038182963897644852432, 5.07982843633272449540368974091, 6.81729297080718468091407044713, 7.13643565053323598192480092714, 7.58125538530298342899767290288, 8.473165188760486697946461654997