| L(s) = 1 | + 3-s + 3·7-s − 2·9-s + 5·11-s − 5·17-s − 19-s + 3·21-s + 23-s − 5·27-s − 9·29-s − 4·31-s + 5·33-s + 3·37-s − 41-s + 3·43-s + 8·47-s + 2·49-s − 5·51-s − 10·53-s − 57-s + 3·59-s + 7·61-s − 6·63-s + 9·67-s + 69-s + 7·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s + 1.50·11-s − 1.21·17-s − 0.229·19-s + 0.654·21-s + 0.208·23-s − 0.962·27-s − 1.67·29-s − 0.718·31-s + 0.870·33-s + 0.493·37-s − 0.156·41-s + 0.457·43-s + 1.16·47-s + 2/7·49-s − 0.700·51-s − 1.37·53-s − 0.132·57-s + 0.390·59-s + 0.896·61-s − 0.755·63-s + 1.09·67-s + 0.120·69-s + 0.830·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.06570713819428, −14.56586924191842, −14.36018319302866, −13.90821432564795, −13.08740040040700, −12.84471000227820, −11.81392121998073, −11.58080918496159, −11.06721249109038, −10.72446453141345, −9.650593522835961, −9.188920893064869, −8.881106950798430, −8.286549113125292, −7.769455590186063, −7.085760207431836, −6.571334264811691, −5.778984281945089, −5.330688249430768, −4.393249611156265, −4.053837567854281, −3.378113466100770, −2.434733606525357, −1.914299211466970, −1.225639349118009, 0,
1.225639349118009, 1.914299211466970, 2.434733606525357, 3.378113466100770, 4.053837567854281, 4.393249611156265, 5.330688249430768, 5.778984281945089, 6.571334264811691, 7.085760207431836, 7.769455590186063, 8.286549113125292, 8.881106950798430, 9.188920893064869, 9.650593522835961, 10.72446453141345, 11.06721249109038, 11.58080918496159, 11.81392121998073, 12.84471000227820, 13.08740040040700, 13.90821432564795, 14.36018319302866, 14.56586924191842, 15.06570713819428
