| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 4·8-s + 3·9-s − 4·10-s + 4·11-s − 6·12-s − 2·13-s − 4·15-s + 5·16-s − 2·17-s − 6·18-s + 4·19-s + 6·20-s − 8·22-s + 2·23-s + 8·24-s − 5·25-s + 4·26-s − 4·27-s − 8·29-s + 8·30-s + 4·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.41·8-s + 9-s − 1.26·10-s + 1.20·11-s − 1.73·12-s − 0.554·13-s − 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s + 0.917·19-s + 1.34·20-s − 1.70·22-s + 0.417·23-s + 1.63·24-s − 25-s + 0.784·26-s − 0.769·27-s − 1.48·29-s + 1.46·30-s + 0.718·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 183 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 139 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 186 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 178 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.74472379500078408925528053213, −7.35555364930301269473262971884, −7.06637445993770824016970542142, −6.87640723027421906109855481571, −6.33162235540943988700356966334, −6.29114639329103663611588379297, −5.71513494212273559894158691685, −5.54433240816924753327134644685, −5.06191074620854839737914845218, −4.80945050074764383963833156064, −4.02420443094422676565869295832, −3.93392533856899540539812742813, −3.20165226337408866605623863049, −2.90946007783184454038611896724, −2.03371527477243769959777579327, −1.97380832758284860844463513314, −1.30153165967718997868510119324, −1.17281374846135084347162539965, 0, 0,
1.17281374846135084347162539965, 1.30153165967718997868510119324, 1.97380832758284860844463513314, 2.03371527477243769959777579327, 2.90946007783184454038611896724, 3.20165226337408866605623863049, 3.93392533856899540539812742813, 4.02420443094422676565869295832, 4.80945050074764383963833156064, 5.06191074620854839737914845218, 5.54433240816924753327134644685, 5.71513494212273559894158691685, 6.29114639329103663611588379297, 6.33162235540943988700356966334, 6.87640723027421906109855481571, 7.06637445993770824016970542142, 7.35555364930301269473262971884, 7.74472379500078408925528053213
