| L(s) = 1 | − 2·3-s + 2·7-s + 9-s − 4·13-s − 4·19-s − 4·21-s + 2·23-s + 4·27-s − 2·29-s + 4·37-s + 8·39-s − 2·41-s − 6·43-s + 6·47-s − 3·49-s − 4·53-s + 8·57-s + 12·59-s + 10·61-s + 2·63-s − 14·67-s − 4·69-s − 8·71-s − 8·73-s + 16·79-s − 11·81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.10·13-s − 0.917·19-s − 0.872·21-s + 0.417·23-s + 0.769·27-s − 0.371·29-s + 0.657·37-s + 1.28·39-s − 0.312·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.549·53-s + 1.05·57-s + 1.56·59-s + 1.28·61-s + 0.251·63-s − 1.71·67-s − 0.481·69-s − 0.949·71-s − 0.936·73-s + 1.80·79-s − 1.22·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.82089307607210, −14.52121672064204, −13.76349255998224, −13.13509321215426, −12.69347162973482, −12.10912207238045, −11.66134630299145, −11.33500403707138, −10.72384911882743, −10.29737471444280, −9.794366951335561, −8.998333165323197, −8.570599661050720, −7.844038588488793, −7.392659938634792, −6.681320089367424, −6.303602380324038, −5.509556876918032, −5.161855263167170, −4.609875189968428, −4.100308051475179, −3.128781123048972, −2.379503845503059, −1.712209608024823, −0.7861143970383193, 0,
0.7861143970383193, 1.712209608024823, 2.379503845503059, 3.128781123048972, 4.100308051475179, 4.609875189968428, 5.161855263167170, 5.509556876918032, 6.303602380324038, 6.681320089367424, 7.392659938634792, 7.844038588488793, 8.570599661050720, 8.998333165323197, 9.794366951335561, 10.29737471444280, 10.72384911882743, 11.33500403707138, 11.66134630299145, 12.10912207238045, 12.69347162973482, 13.13509321215426, 13.76349255998224, 14.52121672064204, 14.82089307607210
