| L(s) = 1 | + i·2-s − i·7-s + i·8-s − 9-s − 11-s + 14-s − 16-s − i·18-s − i·22-s − i·23-s + 29-s + i·37-s − i·43-s + 46-s − 49-s + ⋯ |
| L(s) = 1 | + i·2-s − i·7-s + i·8-s − 9-s − 11-s + 14-s − 16-s − i·18-s − i·22-s − i·23-s + 29-s + i·37-s − i·43-s + 46-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6816688592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6816688592\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 - iT - T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 2iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.51079718904401974351850817746, −12.14444489939289199289128893931, −11.01720550911740402855652502622, −10.29555842557836072172498395314, −8.660735445337753490462854231169, −7.892558870453853779324276599677, −6.90392282848403870644246039249, −5.86088782635053946864062341182, −4.70197917174654042535704225276, −2.78551367007755798973085985436,
2.28332171512070147804322165723, 3.27454930028603520033152893804, 5.16453359959286448991575163680, 6.28445743743243721438375152643, 7.82270203266143885773618920167, 8.948939647607505608978697134519, 9.967080819986157002800541397073, 11.03055908768080276870833170769, 11.70372483322166441461992033855, 12.53910350627727582540120298789
