| L(s) = 1 | − 3-s + 9-s − 4·11-s − 2·13-s + 2·17-s + 4·19-s − 27-s − 10·29-s + 4·33-s − 6·37-s + 2·39-s + 6·41-s + 4·43-s − 8·47-s − 2·51-s − 6·53-s − 4·57-s + 4·59-s + 10·61-s − 4·67-s − 16·71-s − 14·73-s + 8·79-s + 81-s − 4·83-s + 10·87-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.192·27-s − 1.85·29-s + 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 1.16·47-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s + 1.28·61-s − 0.488·67-s − 1.89·71-s − 1.63·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + 1.07·87-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8870738332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8870738332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−15.33098941891754, −14.48804648335965, −14.30075385448038, −13.38239592761556, −12.95682042719332, −12.64746184579894, −11.82994404144711, −11.48990806167501, −10.90892587868767, −10.24713435100103, −9.932720679398737, −9.280699463145042, −8.654156646233612, −7.793589885391396, −7.473821744980563, −7.036826664826946, −6.024088476945736, −5.641880099420281, −5.092060919794430, −4.551483228218842, −3.641927667881794, −3.040261365334698, −2.229370074016685, −1.439581669099816, −0.3768768244647815,
0.3768768244647815, 1.439581669099816, 2.229370074016685, 3.040261365334698, 3.641927667881794, 4.551483228218842, 5.092060919794430, 5.641880099420281, 6.024088476945736, 7.036826664826946, 7.473821744980563, 7.793589885391396, 8.654156646233612, 9.280699463145042, 9.932720679398737, 10.24713435100103, 10.90892587868767, 11.48990806167501, 11.82994404144711, 12.64746184579894, 12.95682042719332, 13.38239592761556, 14.30075385448038, 14.48804648335965, 15.33098941891754
