L(s) = 1 | + 3·3-s − 4·7-s + 6·9-s − 3·11-s − 2·13-s + 3·17-s − 7·19-s − 12·21-s − 6·23-s + 9·27-s + 6·29-s − 10·31-s − 9·33-s − 6·39-s − 11·41-s + 4·43-s + 2·47-s + 9·49-s + 9·51-s − 14·53-s − 21·57-s − 4·59-s + 8·61-s − 24·63-s + 5·67-s − 18·69-s + 2·71-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 1.51·7-s + 2·9-s − 0.904·11-s − 0.554·13-s + 0.727·17-s − 1.60·19-s − 2.61·21-s − 1.25·23-s + 1.73·27-s + 1.11·29-s − 1.79·31-s − 1.56·33-s − 0.960·39-s − 1.71·41-s + 0.609·43-s + 0.291·47-s + 9/7·49-s + 1.26·51-s − 1.92·53-s − 2.78·57-s − 0.520·59-s + 1.02·61-s − 3.02·63-s + 0.610·67-s − 2.16·69-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.226589763444629673853459943284, −7.80616553990494098569552503757, −6.90490229408695066682164139870, −6.26804511259497247397626035236, −5.12135180839156868453261613067, −3.98831218657858314545627224335, −3.41841130884305667674006368738, −2.63801705094948332815391113159, −1.95947456934975683231414172793, 0,
1.95947456934975683231414172793, 2.63801705094948332815391113159, 3.41841130884305667674006368738, 3.98831218657858314545627224335, 5.12135180839156868453261613067, 6.26804511259497247397626035236, 6.90490229408695066682164139870, 7.80616553990494098569552503757, 8.226589763444629673853459943284