| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 2·5-s − 4·6-s + 2·8-s + 7·9-s + 4·10-s − 8·11-s + 2·12-s − 2·13-s − 4·15-s − 4·16-s − 4·17-s − 14·18-s + 4·19-s − 2·20-s + 16·22-s − 8·23-s + 4·24-s + 25-s + 4·26-s + 22·27-s − 8·29-s + 8·30-s + 20·31-s + 2·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s + 0.707·8-s + 7/3·9-s + 1.26·10-s − 2.41·11-s + 0.577·12-s − 0.554·13-s − 1.03·15-s − 16-s − 0.970·17-s − 3.29·18-s + 0.917·19-s − 0.447·20-s + 3.41·22-s − 1.66·23-s + 0.816·24-s + 1/5·25-s + 0.784·26-s + 4.23·27-s − 1.48·29-s + 1.46·30-s + 3.59·31-s + 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7767658721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7767658721\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T - 11 T^{2} - 22 T^{3} + 4 T^{4} - 22 p T^{5} - 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - 10 T^{2} - 32 T^{3} + 115 T^{4} - 32 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $D_{4}$ | \( ( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 18 T + 149 T^{2} - 1242 T^{3} + 10644 T^{4} - 1242 p T^{5} + 149 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 94 T^{2} + 32 T^{3} + 7675 T^{4} + 32 p T^{5} - 94 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 62 T^{2} - 176 T^{3} + 1387 T^{4} - 176 p T^{5} - 62 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 106 T^{2} + 256 T^{3} + 859 T^{4} + 256 p T^{5} + 106 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 16 T + 98 T^{2} + 256 T^{3} + 1747 T^{4} + 256 p T^{5} + 98 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} - 256 T^{3} + 3811 T^{4} - 256 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 8 T + 74 T^{2} + 1408 T^{3} - 12101 T^{4} + 1408 p T^{5} + 74 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.240798109709331312897342160917, −8.089708151240703069566870435198, −7.928213147374756086557247689859, −7.75386058731735483629185610696, −7.46353325770296997942428498316, −7.12647935085304214078127694861, −6.87336783156779508342467461918, −6.85308303657613872306756946993, −6.54955104713719748138276992824, −5.88964673849958571904757387525, −5.81051345741485134007242896699, −5.31389012275215968121546055471, −4.99142644088080065709448021821, −4.75807315389415877424563563496, −4.51313292060716198894049249642, −4.27837659940875134675997898876, −3.92684842738347610951464411642, −3.73218444701199068956290601664, −3.05640747238376356239948802085, −2.80643714367463837841610265120, −2.57070006711520664160635492756, −2.21417347899980003334935144300, −1.73309535914685164203363363044, −1.10051739560853361833487433460, −0.48282261035743114442672166969,
0.48282261035743114442672166969, 1.10051739560853361833487433460, 1.73309535914685164203363363044, 2.21417347899980003334935144300, 2.57070006711520664160635492756, 2.80643714367463837841610265120, 3.05640747238376356239948802085, 3.73218444701199068956290601664, 3.92684842738347610951464411642, 4.27837659940875134675997898876, 4.51313292060716198894049249642, 4.75807315389415877424563563496, 4.99142644088080065709448021821, 5.31389012275215968121546055471, 5.81051345741485134007242896699, 5.88964673849958571904757387525, 6.54955104713719748138276992824, 6.85308303657613872306756946993, 6.87336783156779508342467461918, 7.12647935085304214078127694861, 7.46353325770296997942428498316, 7.75386058731735483629185610696, 7.928213147374756086557247689859, 8.089708151240703069566870435198, 8.240798109709331312897342160917
