| L(s) = 1 | − i·3-s + 2i·7-s − 9-s − i·13-s − 6i·17-s − 2·19-s + 2·21-s + i·27-s + 6·29-s + 2·31-s + 2i·37-s − 39-s − 12·41-s + 4i·43-s + 3·49-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.755i·7-s − 0.333·9-s − 0.277i·13-s − 1.45i·17-s − 0.458·19-s + 0.436·21-s + 0.192i·27-s + 1.11·29-s + 0.359·31-s + 0.328i·37-s − 0.160·39-s − 1.87·41-s + 0.609i·43-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223825119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223825119\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.246338351633735368616140169139, −7.53165228627309250225968932222, −6.68962048994562177913917311785, −6.17292887377036500962934671750, −5.18605028454802285187354187023, −4.67888752918136377747521631973, −3.29640906426465562293523460796, −2.66293715557982258690257735159, −1.68629817374324296152238761595, −0.37211195649736004729770519061,
1.18003313679719617666703663629, 2.35187024738839499729743681622, 3.50090845177034762887597751417, 4.10226210043198031200649336447, 4.79871073124165505586241629979, 5.74835771598664034187146743665, 6.51625251273592792889649360143, 7.16816886494807905424675747520, 8.223191789834371459505444545092, 8.553393549374940667645947902979
