| L(s) = 1 | − 5-s − 3·7-s + 5·13-s + 5·17-s + 19-s − 4·23-s + 25-s − 5·29-s − 2·31-s + 3·35-s + 3·37-s + 2·41-s + 2·43-s + 6·47-s + 2·49-s + 4·59-s − 8·61-s − 5·65-s − 6·67-s + 3·71-s + 10·73-s − 10·79-s − 9·83-s − 5·85-s + 12·89-s − 15·91-s − 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.38·13-s + 1.21·17-s + 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.928·29-s − 0.359·31-s + 0.507·35-s + 0.493·37-s + 0.312·41-s + 0.304·43-s + 0.875·47-s + 2/7·49-s + 0.520·59-s − 1.02·61-s − 0.620·65-s − 0.733·67-s + 0.356·71-s + 1.17·73-s − 1.12·79-s − 0.987·83-s − 0.542·85-s + 1.27·89-s − 1.57·91-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.05884812496676, −13.66858945529243, −13.06421477992544, −12.74943632480303, −12.17087513443830, −11.77890471934286, −11.10916204739154, −10.75084155921547, −10.12862188051855, −9.636904225791159, −9.230091272814426, −8.608852324917584, −8.107781119282476, −7.510160532590540, −7.143764384162019, −6.343906656362119, −5.932036468238867, −5.621729713027470, −4.749425217427834, −3.935381251812805, −3.663849643842154, −3.159667865587244, −2.447626994581324, −1.520185582967745, −0.8569309386162201, 0,
0.8569309386162201, 1.520185582967745, 2.447626994581324, 3.159667865587244, 3.663849643842154, 3.935381251812805, 4.749425217427834, 5.621729713027470, 5.932036468238867, 6.343906656362119, 7.143764384162019, 7.510160532590540, 8.107781119282476, 8.608852324917584, 9.230091272814426, 9.636904225791159, 10.12862188051855, 10.75084155921547, 11.10916204739154, 11.77890471934286, 12.17087513443830, 12.74943632480303, 13.06421477992544, 13.66858945529243, 14.05884812496676
