| L(s) = 1 | + 2·11-s − 3i·17-s + 19-s + 3i·23-s − 4·29-s − 5·31-s − 10i·37-s + 6·41-s − 6i·43-s − 8i·47-s + 7·49-s + 3i·53-s + 5·61-s + 2i·67-s + 2·71-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 0.727i·17-s + 0.229·19-s + 0.625i·23-s − 0.742·29-s − 0.898·31-s − 1.64i·37-s + 0.937·41-s − 0.914i·43-s − 1.16i·47-s + 49-s + 0.412i·53-s + 0.640·61-s + 0.244i·67-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.715571311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.715571311\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−7.976411683820534792778389981786, −7.21885384900375487185320826446, −6.84611206779404725206113332739, −5.60246499593008097503309344670, −5.44899571892131381918563815503, −4.16571129256246658456308924274, −3.70717818248219795825376849036, −2.62506064172105849044173494636, −1.72382265634785801797966192592, −0.51194152760306697267346069245,
1.02795872785757080851385569456, 2.00252257611571403326738883369, 3.03506026265434456505779216681, 3.87652428614993324478657608781, 4.56320977329341254096274747923, 5.46513242743369861100795806153, 6.21740941516161929256826489364, 6.78405464030845245417259376330, 7.67419059854830156854979116121, 8.225581177931808512771815524367
