L(s) = 1 | − 3-s + 2·7-s + 9-s + 11-s − 2·13-s + 2·19-s − 2·21-s − 23-s − 5·25-s − 27-s + 10·29-s − 4·31-s − 33-s − 2·37-s + 2·39-s − 2·41-s + 2·43-s + 8·47-s − 3·49-s + 4·53-s − 2·57-s − 12·59-s + 6·61-s + 2·63-s + 2·67-s + 69-s − 6·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.458·19-s − 0.436·21-s − 0.208·23-s − 25-s − 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.174·33-s − 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.549·53-s − 0.264·57-s − 1.56·59-s + 0.768·61-s + 0.251·63-s + 0.244·67-s + 0.120·69-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−14.84791376154008, −14.17548292985151, −13.91134220170498, −13.35163564697232, −12.51762015654480, −12.23198744447111, −11.70595284861862, −11.33090388288238, −10.66065146687303, −10.22471508081074, −9.700886577819702, −9.074717984295478, −8.510743617538064, −7.835033521973855, −7.478511382501547, −6.770880687411833, −6.286043701346632, −5.522853948759514, −5.184894152123467, −4.445601793108554, −4.050206187336432, −3.166357878043828, −2.409057520321815, −1.663819773765762, −0.9884226079285945, 0,
0.9884226079285945, 1.663819773765762, 2.409057520321815, 3.166357878043828, 4.050206187336432, 4.445601793108554, 5.184894152123467, 5.522853948759514, 6.286043701346632, 6.770880687411833, 7.478511382501547, 7.835033521973855, 8.510743617538064, 9.074717984295478, 9.700886577819702, 10.22471508081074, 10.66065146687303, 11.33090388288238, 11.70595284861862, 12.23198744447111, 12.51762015654480, 13.35163564697232, 13.91134220170498, 14.17548292985151, 14.84791376154008