| L(s) = 1 | + 4i·7-s + 6i·13-s + 2i·17-s − 4·19-s − 8i·23-s − 6·29-s − 6i·37-s − 10·41-s + 4i·43-s − 8i·47-s − 9·49-s + 10i·53-s + 6·61-s − 4i·67-s + 14i·73-s + ⋯ |
| L(s) = 1 | + 1.51i·7-s + 1.66i·13-s + 0.485i·17-s − 0.917·19-s − 1.66i·23-s − 1.11·29-s − 0.986i·37-s − 1.56·41-s + 0.609i·43-s − 1.16i·47-s − 1.28·49-s + 1.37i·53-s + 0.768·61-s − 0.488i·67-s + 1.63i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9091538250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9091538250\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 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
−9.410822026805819413494941263195, −8.739701077698961605220149992299, −8.397014709772719297954035273539, −7.07621718091147669727850792292, −6.39062752872651967200390688600, −5.67779974714478237673948636374, −4.68509208498912455801695255453, −3.85701014085529805346463870366, −2.47971382103654249364889919860, −1.87289842262772597939436652973,
0.32308798839363023266101151492, 1.58499580014691248839901629236, 3.14479890968177729318094330638, 3.78032444591387555605570507771, 4.84673051643619427962800530776, 5.64021483342534048799871447151, 6.67702071371587925362767537663, 7.49167269431350425143835028422, 7.926146970154634680894983594380, 8.920615677806803358136250301106
