| L(s) = 1 | + 3·7-s + 11-s − 13-s + 3·17-s − 4·19-s − 2·23-s − 3·29-s + 5·31-s − 2·37-s − 2·43-s − 9·47-s + 2·49-s + 5·53-s + 7·59-s − 11·61-s − 3·67-s − 8·71-s − 2·73-s + 3·77-s + 10·79-s + 3·83-s + 8·89-s − 3·91-s + 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.301·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s − 0.417·23-s − 0.557·29-s + 0.898·31-s − 0.328·37-s − 0.304·43-s − 1.31·47-s + 2/7·49-s + 0.686·53-s + 0.911·59-s − 1.40·61-s − 0.366·67-s − 0.949·71-s − 0.234·73-s + 0.341·77-s + 1.12·79-s + 0.329·83-s + 0.847·89-s − 0.314·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.80080056373051, −14.45568158259157, −13.90215079713240, −13.35061568946060, −12.81945019321725, −12.11179354478142, −11.81419018198068, −11.31895656281951, −10.67676446991125, −10.24457096698522, −9.703526311917178, −8.982900658311017, −8.535756707948885, −7.898596790746452, −7.644476673816450, −6.825043361733254, −6.285609488675198, −5.663908870878294, −4.975784546739865, −4.583617126732598, −3.891526240069369, −3.236646019417535, −2.363883517306970, −1.761560250688124, −1.100377956872247, 0,
1.100377956872247, 1.761560250688124, 2.363883517306970, 3.236646019417535, 3.891526240069369, 4.583617126732598, 4.975784546739865, 5.663908870878294, 6.285609488675198, 6.825043361733254, 7.644476673816450, 7.898596790746452, 8.535756707948885, 8.982900658311017, 9.703526311917178, 10.24457096698522, 10.67676446991125, 11.31895656281951, 11.81419018198068, 12.11179354478142, 12.81945019321725, 13.35061568946060, 13.90215079713240, 14.45568158259157, 14.80080056373051
