| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 2·5-s + 4·6-s − 2·7-s + 2·8-s + 9-s − 4·10-s + 8·11-s − 2·12-s + 14·13-s + 4·14-s − 4·15-s − 4·16-s − 6·17-s − 2·18-s + 2·19-s + 2·20-s + 4·21-s − 16·22-s + 3·23-s − 4·24-s + 11·25-s − 28·26-s + 2·27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s + 0.707·8-s + 1/3·9-s − 1.26·10-s + 2.41·11-s − 0.577·12-s + 3.88·13-s + 1.06·14-s − 1.03·15-s − 16-s − 1.45·17-s − 0.471·18-s + 0.458·19-s + 0.447·20-s + 0.872·21-s − 3.41·22-s + 0.625·23-s − 0.816·24-s + 11/5·25-s − 5.49·26-s + 0.384·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.421146971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421146971\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T + 22 T^{2} + 112 T^{3} - 113 T^{4} + 112 p T^{5} + 22 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - 25 T^{2} + 36 T^{3} + 420 T^{4} + 36 p T^{5} - 25 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 9 T + 17 T^{2} + 54 T^{3} + 906 T^{4} + 54 p T^{5} + 17 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 7 T - 23 T^{2} + 14 T^{3} + 2002 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 9 T - 7 T^{2} + 54 T^{3} + 2742 T^{4} + 54 p T^{5} - 7 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 47 | $D_{4}$ | \( ( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 65 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - T + 11 T^{2} + 128 T^{3} - 3440 T^{4} + 128 p T^{5} + 11 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 13 T + 7 T^{2} - 364 T^{3} + 10432 T^{4} - 364 p T^{5} + 7 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 3 T - 121 T^{2} - 36 T^{3} + 11220 T^{4} - 36 p T^{5} - 121 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 19 T + 222 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 9 T - 103 T^{2} + 54 T^{3} + 18726 T^{4} + 54 p T^{5} - 103 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 14 T + 10 T^{2} - 112 T^{3} + 6175 T^{4} - 112 p T^{5} + 10 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.144325962756864746498641701013, −7.38474402246854313751929396183, −7.23973489423363799615641590840, −6.93684760129807240353044800478, −6.78993651485837534191489007827, −6.42328422609142078290890984412, −6.39246210159131191168886133129, −6.36180696536628095206601972150, −6.09921726721841033672921502160, −5.53192785511085364996752855248, −5.42282855183347354190328533060, −5.39608951024021244020050534097, −4.79920863205867065813416041662, −4.55626558573150551997766725923, −3.99861042369625814833989955675, −3.91698530874514036110549935709, −3.80605076785381613302840167360, −3.58636921122831018395502785470, −2.90541959188312581433680185504, −2.68899801264906544670085670522, −2.12285541535609941235298984318, −1.62184992632624417131358898284, −1.20874050595958855215314506543, −0.972198847389114395569722339515, −0.77980177529961547556015709261,
0.77980177529961547556015709261, 0.972198847389114395569722339515, 1.20874050595958855215314506543, 1.62184992632624417131358898284, 2.12285541535609941235298984318, 2.68899801264906544670085670522, 2.90541959188312581433680185504, 3.58636921122831018395502785470, 3.80605076785381613302840167360, 3.91698530874514036110549935709, 3.99861042369625814833989955675, 4.55626558573150551997766725923, 4.79920863205867065813416041662, 5.39608951024021244020050534097, 5.42282855183347354190328533060, 5.53192785511085364996752855248, 6.09921726721841033672921502160, 6.36180696536628095206601972150, 6.39246210159131191168886133129, 6.42328422609142078290890984412, 6.78993651485837534191489007827, 6.93684760129807240353044800478, 7.23973489423363799615641590840, 7.38474402246854313751929396183, 8.144325962756864746498641701013
