| L(s) = 1 | − 2·3-s + 4·5-s + 3·9-s + 4·11-s − 8·15-s + 4·17-s − 8·19-s − 4·23-s + 4·25-s − 4·27-s + 16·31-s − 8·33-s + 8·37-s + 4·41-s − 16·43-s + 12·45-s + 8·47-s − 8·51-s + 4·53-s + 16·55-s + 16·57-s − 16·59-s + 8·61-s − 16·67-s + 8·69-s − 4·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s − 2.06·15-s + 0.970·17-s − 1.83·19-s − 0.834·23-s + 4/5·25-s − 0.769·27-s + 2.87·31-s − 1.39·33-s + 1.31·37-s + 0.624·41-s − 2.43·43-s + 1.78·45-s + 1.16·47-s − 1.12·51-s + 0.549·53-s + 2.15·55-s + 2.11·57-s − 2.08·59-s + 1.02·61-s − 1.95·67-s + 0.963·69-s − 0.474·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.047588157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.047588157\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 16 T + 118 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 112 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−7.68742066106454099519409554696, −7.59754952493918638532494705160, −6.98749367846939955155565705517, −6.62742805603042708442395695884, −6.20615407374344862714351527046, −6.19337992088851168134332544181, −5.98701100628049734013957454183, −5.79716131908504674783806117743, −4.96802595292866187351412874698, −4.93439455815952697113790098947, −4.45433284318597008617127379976, −4.24688682768405466642746601112, −3.63303687257312336016518181841, −3.35061619128723922666464315328, −2.58501180612700446764246689877, −2.30101040335878366305906454952, −1.92645126436223169599472639519, −1.37358040504496856404541466744, −1.10102610468143054209071671379, −0.45180679034758887988236174591,
0.45180679034758887988236174591, 1.10102610468143054209071671379, 1.37358040504496856404541466744, 1.92645126436223169599472639519, 2.30101040335878366305906454952, 2.58501180612700446764246689877, 3.35061619128723922666464315328, 3.63303687257312336016518181841, 4.24688682768405466642746601112, 4.45433284318597008617127379976, 4.93439455815952697113790098947, 4.96802595292866187351412874698, 5.79716131908504674783806117743, 5.98701100628049734013957454183, 6.19337992088851168134332544181, 6.20615407374344862714351527046, 6.62742805603042708442395695884, 6.98749367846939955155565705517, 7.59754952493918638532494705160, 7.68742066106454099519409554696
