| L(s) = 1 | − 2·13-s + 2·17-s + 2·19-s − 8·23-s + 8·29-s + 4·31-s + 6·37-s + 10·41-s + 2·43-s − 6·47-s + 6·53-s − 12·59-s − 2·61-s − 14·67-s + 6·71-s − 10·73-s − 4·79-s + 12·83-s − 14·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.554·13-s + 0.485·17-s + 0.458·19-s − 1.66·23-s + 1.48·29-s + 0.718·31-s + 0.986·37-s + 1.56·41-s + 0.304·43-s − 0.875·47-s + 0.824·53-s − 1.56·59-s − 0.256·61-s − 1.71·67-s + 0.712·71-s − 1.17·73-s − 0.450·79-s + 1.31·83-s − 1.48·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.57916662040327, −12.85708681740211, −12.36847960851134, −12.05989213739572, −11.64673716439251, −11.09452315265508, −10.46261951760843, −10.06861708163972, −9.738813493952104, −9.172102705061952, −8.634290623492706, −8.020621085925090, −7.687760736764252, −7.297665949153458, −6.480290470211581, −6.100175845398691, −5.725733872287106, −4.942389063303546, −4.483409113908038, −4.087765443602945, −3.282241919830289, −2.763632488110863, −2.285102995669500, −1.452594190900314, −0.8528748065596784, 0,
0.8528748065596784, 1.452594190900314, 2.285102995669500, 2.763632488110863, 3.282241919830289, 4.087765443602945, 4.483409113908038, 4.942389063303546, 5.725733872287106, 6.100175845398691, 6.480290470211581, 7.297665949153458, 7.687760736764252, 8.020621085925090, 8.634290623492706, 9.172102705061952, 9.738813493952104, 10.06861708163972, 10.46261951760843, 11.09452315265508, 11.64673716439251, 12.05989213739572, 12.36847960851134, 12.85708681740211, 13.57916662040327
