| L(s) = 1 | + 5·7-s − 2·11-s + 13-s − 3·17-s + 2·19-s − 4·23-s + 6·29-s + 4·31-s − 11·37-s − 8·41-s − 43-s − 9·47-s + 18·49-s − 12·53-s + 6·59-s + 6·67-s + 7·71-s + 2·73-s − 10·77-s − 12·79-s + 16·83-s + 10·89-s + 5·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 0.603·11-s + 0.277·13-s − 0.727·17-s + 0.458·19-s − 0.834·23-s + 1.11·29-s + 0.718·31-s − 1.80·37-s − 1.24·41-s − 0.152·43-s − 1.31·47-s + 18/7·49-s − 1.64·53-s + 0.781·59-s + 0.733·67-s + 0.830·71-s + 0.234·73-s − 1.13·77-s − 1.35·79-s + 1.75·83-s + 1.05·89-s + 0.524·91-s + 1.01·97-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 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.79478648170777, −14.33195045717523, −13.73447367695103, −13.62991125713345, −12.78496473975748, −12.09407673708196, −11.75871035839701, −11.29071362688701, −10.69936739400767, −10.32945722040827, −9.733585833829487, −8.879629747824380, −8.443299434878383, −8.012781897423723, −7.684933321935873, −6.731928562711734, −6.455654743583355, −5.384326792428555, −5.098094986789525, −4.643496197218301, −3.930347547372792, −3.187977620721909, −2.320932940637962, −1.776661390065745, −1.139997619740426, 0,
1.139997619740426, 1.776661390065745, 2.320932940637962, 3.187977620721909, 3.930347547372792, 4.643496197218301, 5.098094986789525, 5.384326792428555, 6.455654743583355, 6.731928562711734, 7.684933321935873, 8.012781897423723, 8.443299434878383, 8.879629747824380, 9.733585833829487, 10.32945722040827, 10.69936739400767, 11.29071362688701, 11.75871035839701, 12.09407673708196, 12.78496473975748, 13.62991125713345, 13.73447367695103, 14.33195045717523, 14.79478648170777
