| L(s) = 1 | − 3·2-s + 4·4-s − 9·8-s − 18·9-s + 27·16-s + 54·18-s + 50·25-s + 216·29-s − 36·32-s − 72·36-s − 76·37-s − 150·50-s + 12·53-s − 648·58-s + 17·64-s + 162·72-s + 228·74-s + 81·81-s + 200·100-s − 36·106-s + 212·109-s − 888·113-s + 864·116-s + 206·121-s + 127-s − 51·128-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·2-s + 4-s − 9/8·8-s − 2·9-s + 1.68·16-s + 3·18-s + 2·25-s + 7.44·29-s − 9/8·32-s − 2·36-s − 2.05·37-s − 3·50-s + 0.226·53-s − 11.1·58-s + 0.265·64-s + 9/4·72-s + 3.08·74-s + 81-s + 2·100-s − 0.339·106-s + 1.94·109-s − 7.85·113-s + 7.44·116-s + 1.70·121-s + 0.00787·127-s − 0.398·128-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6433865306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6433865306\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 85 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )( 1 + 6 T - 85 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T - 205 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )( 1 + 18 T - 205 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 38 T + 75 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2}( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T - 2773 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 118 T + 9435 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} )( 1 + 118 T + 9435 T^{2} + 118 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - 114 T + p^{2} T^{2} )^{2}( 1 + 114 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 94 T + 2595 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} )( 1 + 94 T + 2595 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
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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
−8.809738774431756866311051940981, −8.595772507173489990894419901315, −8.522354181547407865765497086104, −8.325208226323375168916873677555, −8.027517419785100981296416614497, −7.85601521842932679072192884807, −7.28631364179980387682869361479, −6.96167453042350617797430890944, −6.63124018515412501371640011327, −6.41622393065800592886244053649, −6.32569093789429150952913399032, −6.14479553811743394202168514220, −5.34104129833482215996011120951, −5.18720802501965687094292406129, −5.13725417330531090305023791163, −4.58241377303613680724560675371, −4.32697336905682409801135281090, −3.66297738625382555783425172414, −3.12080309453289138446010568164, −2.87622842334765462106938250509, −2.68971693545973588849000237146, −2.54318815275178338633218991284, −1.25260403614076374882492227256, −1.07062466544909883439730726583, −0.39537075190574154043995249685,
0.39537075190574154043995249685, 1.07062466544909883439730726583, 1.25260403614076374882492227256, 2.54318815275178338633218991284, 2.68971693545973588849000237146, 2.87622842334765462106938250509, 3.12080309453289138446010568164, 3.66297738625382555783425172414, 4.32697336905682409801135281090, 4.58241377303613680724560675371, 5.13725417330531090305023791163, 5.18720802501965687094292406129, 5.34104129833482215996011120951, 6.14479553811743394202168514220, 6.32569093789429150952913399032, 6.41622393065800592886244053649, 6.63124018515412501371640011327, 6.96167453042350617797430890944, 7.28631364179980387682869361479, 7.85601521842932679072192884807, 8.027517419785100981296416614497, 8.325208226323375168916873677555, 8.522354181547407865765497086104, 8.595772507173489990894419901315, 8.809738774431756866311051940981
