| L(s) = 1 | − 5.16i·7-s − 3.05i·11-s + 0.837i·13-s − 6.32·19-s − 4.47·23-s + 11.1i·37-s − 10.3i·41-s + 2.82·47-s − 19.6·49-s − 5.65·53-s − 5.42i·59-s − 15.7·77-s + 18.8i·89-s + 4.32·91-s + 17.1i·103-s + ⋯ |
| L(s) = 1 | − 1.95i·7-s − 0.921i·11-s + 0.232i·13-s − 1.45·19-s − 0.932·23-s + 1.83i·37-s − 1.61i·41-s + 0.412·47-s − 2.80·49-s − 0.777·53-s − 0.706i·59-s − 1.79·77-s + 1.99i·89-s + 0.453·91-s + 1.69i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2076353743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2076353743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5.16iT - 7T^{2} \) |
| 11 | \( 1 + 3.05iT - 11T^{2} \) |
| 13 | \( 1 - 0.837iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 11.1iT - 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 5.42iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 18.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.53125564770632930100187065229, −6.60272735010745222420820528784, −6.40639006428407800035960937436, −5.29399632909694818101743961266, −4.38766370834751807791351292355, −3.93173710251363081910113390073, −3.22370694792292292874337684844, −2.01806094684442008562547345702, −0.999927028776667244085948464983, −0.05278413464487421641645368001,
1.81066884892778001002016403329, 2.27474615835357897050585023310, 3.08924343805673315773492996741, 4.22766378906136468603886536285, 4.82609656960892389601410219917, 5.79638391565113384841622298013, 6.04075574610854152758547537001, 6.92925397518104936597984205457, 7.81690674733328910007701521029, 8.425548065721254266431759353701
