| L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s − 4·11-s + 2·13-s + 4·17-s − 4·21-s + 8·23-s − 6·27-s + 10·29-s + 6·31-s + 8·33-s − 6·37-s − 4·39-s + 14·41-s + 8·43-s − 4·47-s + 3·49-s − 8·51-s − 8·53-s − 10·59-s − 6·61-s + 4·63-s − 4·67-s − 16·69-s − 4·71-s + 22·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 2/3·9-s − 1.20·11-s + 0.554·13-s + 0.970·17-s − 0.872·21-s + 1.66·23-s − 1.15·27-s + 1.85·29-s + 1.07·31-s + 1.39·33-s − 0.986·37-s − 0.640·39-s + 2.18·41-s + 1.21·43-s − 0.583·47-s + 3/7·49-s − 1.12·51-s − 1.09·53-s − 1.30·59-s − 0.768·61-s + 0.503·63-s − 0.488·67-s − 1.92·69-s − 0.474·71-s + 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.140712079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.140712079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 133 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 101 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 262 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 122 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 198 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.997817766239093368584896637960, −8.568704442062272937700776691322, −8.113525760896149142941644389840, −7.71636946585161733462585470812, −7.64997614619413494519724505227, −7.15871599984143801493225201712, −6.45374957740923811960333249577, −6.40248591351541448346145230388, −5.72578583867714300749631665858, −5.70645458785488601107201515644, −4.93974897452160256386585991469, −4.85763268688068697215320293662, −4.57977576161775762894777885259, −3.90976902771277910202252155678, −3.22105487340315837040061081294, −2.98081772993730869873327434177, −2.34703569094201809767295297023, −1.69974817186533828264479277417, −0.932723821089314837512743804676, −0.67762939405580346289441619077,
0.67762939405580346289441619077, 0.932723821089314837512743804676, 1.69974817186533828264479277417, 2.34703569094201809767295297023, 2.98081772993730869873327434177, 3.22105487340315837040061081294, 3.90976902771277910202252155678, 4.57977576161775762894777885259, 4.85763268688068697215320293662, 4.93974897452160256386585991469, 5.70645458785488601107201515644, 5.72578583867714300749631665858, 6.40248591351541448346145230388, 6.45374957740923811960333249577, 7.15871599984143801493225201712, 7.64997614619413494519724505227, 7.71636946585161733462585470812, 8.113525760896149142941644389840, 8.568704442062272937700776691322, 8.997817766239093368584896637960
