L(s) = 1 | + i·7-s − 2·11-s − 6i·13-s − 4i·17-s + 4·19-s − 2i·23-s − 2·29-s − 2i·37-s − 4i·43-s + 12i·47-s − 49-s + 6i·53-s − 8·59-s + 6·61-s + 8i·67-s + ⋯ |
L(s) = 1 | + 0.377i·7-s − 0.603·11-s − 1.66i·13-s − 0.970i·17-s + 0.917·19-s − 0.417i·23-s − 0.371·29-s − 0.328i·37-s − 0.609i·43-s + 1.75i·47-s − 0.142·49-s + 0.824i·53-s − 1.04·59-s + 0.768·61-s + 0.977i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7101858135\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7101858135\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−7.63254544422784445227909246295, −7.29927822443569644251915368845, −6.17524548117577484452384575213, −5.50892203225646806213486388173, −5.07829159415646891310801848433, −4.09716352807958767177191724275, −2.87599718117033692806891964981, −2.79409807269288405723341714778, −1.29062352290121839748081818876, −0.18044265537975429540259445565,
1.35607461386975175965578174381, 2.11402219007738449201948800389, 3.24612504888222370662574153703, 3.96974842120990090346244940649, 4.70227468616556131251141092364, 5.48520834236951990730073215459, 6.28503661962175324389174650315, 6.98257935910849707866313867302, 7.56156523973098523448654439858, 8.331916542968755470370379987275