| L(s) = 1 | − 2·7-s + 11-s + 5·13-s + 19-s + 3·23-s + 3·29-s + 31-s + 2·37-s + 6·41-s − 11·43-s − 12·47-s − 3·49-s − 12·53-s + 6·59-s + 2·61-s − 2·67-s − 9·71-s + 2·73-s − 2·77-s − 8·79-s + 9·83-s + 3·89-s − 10·91-s − 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 1.38·13-s + 0.229·19-s + 0.625·23-s + 0.557·29-s + 0.179·31-s + 0.328·37-s + 0.937·41-s − 1.67·43-s − 1.75·47-s − 3/7·49-s − 1.64·53-s + 0.781·59-s + 0.256·61-s − 0.244·67-s − 1.06·71-s + 0.234·73-s − 0.227·77-s − 0.900·79-s + 0.987·83-s + 0.317·89-s − 1.04·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.01789389480828, −14.56811717674784, −13.97350888623229, −13.41073045732625, −12.97824985727130, −12.67590842098436, −11.77790128804588, −11.43678341385720, −10.95077954490743, −10.21127956544847, −9.848689512517499, −9.175207041216032, −8.747365586547277, −8.134222241562585, −7.613257025743210, −6.685751310700400, −6.474149999440293, −5.955891865632297, −5.123828177921222, −4.587562336915843, −3.729042712235126, −3.318241352283076, −2.717776156967517, −1.649140320517024, −1.063826874496401, 0,
1.063826874496401, 1.649140320517024, 2.717776156967517, 3.318241352283076, 3.729042712235126, 4.587562336915843, 5.123828177921222, 5.955891865632297, 6.474149999440293, 6.685751310700400, 7.613257025743210, 8.134222241562585, 8.747365586547277, 9.175207041216032, 9.848689512517499, 10.21127956544847, 10.95077954490743, 11.43678341385720, 11.77790128804588, 12.67590842098436, 12.97824985727130, 13.41073045732625, 13.97350888623229, 14.56811717674784, 15.01789389480828
