| L(s) = 1 | + 7-s + 2·11-s + 2·13-s − 4·17-s + 8·23-s − 5·29-s + 8·31-s − 6·37-s − 12·41-s − 7·43-s − 12·47-s − 6·49-s − 7·53-s + 3·59-s + 8·61-s + 67-s − 12·71-s − 10·73-s + 2·77-s + 10·79-s + 12·83-s + 15·89-s + 2·91-s + 5·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s + 0.554·13-s − 0.970·17-s + 1.66·23-s − 0.928·29-s + 1.43·31-s − 0.986·37-s − 1.87·41-s − 1.06·43-s − 1.75·47-s − 6/7·49-s − 0.961·53-s + 0.390·59-s + 1.02·61-s + 0.122·67-s − 1.42·71-s − 1.17·73-s + 0.227·77-s + 1.12·79-s + 1.31·83-s + 1.58·89-s + 0.209·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.05661202313265, −12.92830905156446, −11.96247604515480, −11.67832852539409, −11.40466094691872, −10.80559031168208, −10.43393354842217, −9.820893241840757, −9.391482820436704, −8.756875450186824, −8.560286165541733, −8.069406705311328, −7.407460452142916, −6.799426008251749, −6.559590274079866, −6.125965677206233, −5.226904512425576, −4.904011912595208, −4.569552401416285, −3.684750180725031, −3.366495797740786, −2.815716452277791, −1.808376773673850, −1.675635299720272, −0.8284542350996868, 0,
0.8284542350996868, 1.675635299720272, 1.808376773673850, 2.815716452277791, 3.366495797740786, 3.684750180725031, 4.569552401416285, 4.904011912595208, 5.226904512425576, 6.125965677206233, 6.559590274079866, 6.799426008251749, 7.407460452142916, 8.069406705311328, 8.560286165541733, 8.756875450186824, 9.391482820436704, 9.820893241840757, 10.43393354842217, 10.80559031168208, 11.40466094691872, 11.67832852539409, 11.96247604515480, 12.92830905156446, 13.05661202313265
