| L(s) = 1 | + i·3-s + 5-s − 0.968·7-s − 9-s + (3.16 + i)11-s + 4.13i·13-s + i·15-s − 0.968i·17-s + 5.09·19-s − 0.968i·21-s − 7.06i·23-s + 25-s − i·27-s + 3.16i·29-s + (−1 + 3.16i)33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447·5-s − 0.366·7-s − 0.333·9-s + (0.953 + 0.301i)11-s + 1.14i·13-s + 0.258i·15-s − 0.234i·17-s + 1.16·19-s − 0.211i·21-s − 1.47i·23-s + 0.200·25-s − 0.192i·27-s + 0.587i·29-s + (−0.174 + 0.550i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.976871349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976871349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + (-3.16 - i)T \) |
| good | 7 | \( 1 + 0.968T + 7T^{2} \) |
| 13 | \( 1 - 4.13iT - 13T^{2} \) |
| 17 | \( 1 + 0.968iT - 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 23 | \( 1 + 7.06iT - 23T^{2} \) |
| 29 | \( 1 - 3.16iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 9.06T + 37T^{2} \) |
| 41 | \( 1 - 11.4iT - 41T^{2} \) |
| 43 | \( 1 + 0.968T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 - 3.06iT - 59T^{2} \) |
| 61 | \( 1 - 4.38iT - 61T^{2} \) |
| 67 | \( 1 - 5.06iT - 67T^{2} \) |
| 71 | \( 1 - 9.06iT - 71T^{2} \) |
| 73 | \( 1 - 0.257iT - 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 9.06T + 97T^{2} \) |
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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
−9.127792044691086116937555926994, −8.517882580717967235673553634179, −7.37752204309825200940853921506, −6.59256596152431159720654707563, −6.07222511561800039706921380205, −4.92590766264386205524245026430, −4.35504448385580465885560137157, −3.38763652850889313629775705886, −2.41727817557103050911643753240, −1.17468005716145731613494875714,
0.74895398931156652786746570772, 1.76759714115748091924669606319, 3.01521114200887218548432875308, 3.63707373255549921050933749270, 4.94679143032949224121488525706, 5.88015422119776261237570257512, 6.20209022175637510859180390333, 7.33099908521649964242788999468, 7.77192778444578472799886832976, 8.732126010039124222771402956322
