| L(s) = 1 | + 4·7-s + 2·11-s + 6·13-s − 3·17-s + 19-s − 4·23-s + 8·29-s − 5·31-s + 8·37-s − 12·41-s − 7·43-s + 2·47-s + 9·49-s − 11·53-s − 4·59-s − 6·61-s + 13·67-s − 5·71-s + 2·73-s + 8·77-s − 79-s + 6·89-s + 24·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.603·11-s + 1.66·13-s − 0.727·17-s + 0.229·19-s − 0.834·23-s + 1.48·29-s − 0.898·31-s + 1.31·37-s − 1.87·41-s − 1.06·43-s + 0.291·47-s + 9/7·49-s − 1.51·53-s − 0.520·59-s − 0.768·61-s + 1.58·67-s − 0.593·71-s + 0.234·73-s + 0.911·77-s − 0.112·79-s + 0.635·89-s + 2.51·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 331 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.00127856404170, −12.34845524590802, −11.88180360981631, −11.47860638073168, −11.19490243755959, −10.71508546039672, −10.34081768030110, −9.652241185611334, −9.171302465321269, −8.643719759931410, −8.221128673810803, −8.083600619827537, −7.409747362140363, −6.686435124515123, −6.405565439345754, −5.922259495990662, −5.255087088377159, −4.782705466419200, −4.371373874330326, −3.786486692265124, −3.366118689672478, −2.581459468105913, −1.855739501339072, −1.466192677755970, −1.018891496883321, 0,
1.018891496883321, 1.466192677755970, 1.855739501339072, 2.581459468105913, 3.366118689672478, 3.786486692265124, 4.371373874330326, 4.782705466419200, 5.255087088377159, 5.922259495990662, 6.405565439345754, 6.686435124515123, 7.409747362140363, 8.083600619827537, 8.221128673810803, 8.643719759931410, 9.171302465321269, 9.652241185611334, 10.34081768030110, 10.71508546039672, 11.19490243755959, 11.47860638073168, 11.88180360981631, 12.34845524590802, 13.00127856404170
