| L(s) = 1 | + i·2-s − 4-s − 5-s − i·8-s − i·10-s − 3.98i·11-s + 0.0681i·13-s + 16-s − 7.32·17-s − 2.03i·19-s + 20-s + 3.98·22-s + 3.73i·23-s + 25-s − 0.0681·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.447·5-s − 0.353i·8-s − 0.316i·10-s − 1.20i·11-s + 0.0189i·13-s + 0.250·16-s − 1.77·17-s − 0.466i·19-s + 0.223·20-s + 0.848·22-s + 0.778i·23-s + 0.200·25-s − 0.0133·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015552909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015552909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3.98iT - 11T^{2} \) |
| 13 | \( 1 - 0.0681iT - 13T^{2} \) |
| 17 | \( 1 + 7.32T + 17T^{2} \) |
| 19 | \( 1 + 2.03iT - 19T^{2} \) |
| 23 | \( 1 - 3.73iT - 23T^{2} \) |
| 29 | \( 1 - 0.898iT - 29T^{2} \) |
| 31 | \( 1 - 4.82iT - 31T^{2} \) |
| 37 | \( 1 + 4.06T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 + 0.964T + 43T^{2} \) |
| 47 | \( 1 - 1.66T + 47T^{2} \) |
| 53 | \( 1 + 13.2iT - 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.52iT - 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 9.93iT - 71T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 1.82T + 89T^{2} \) |
| 97 | \( 1 - 17.1iT - 97T^{2} \) |
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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
−8.653689772884540546240245999955, −7.88755598498535857015685623309, −6.87999806740020162807819989518, −6.66792582576333683324891485612, −5.59299425270837090488699786902, −5.01848623684712599065557448648, −4.05937262483848982571384110353, −3.40803904057681810139172114896, −2.29335129497667185311679317782, −0.813826627751919386180422449043,
0.37975000240259758578736527255, 1.86648869019899120402473907624, 2.45379329318387443905665135299, 3.59461821312767347261588258154, 4.42056006634038559524431600597, 4.76910273312095498223697773788, 5.94998683682888345090776165398, 6.77913632596748129377742356535, 7.45650070891260288175011395153, 8.277075078628456684903285650115
