| L(s) = 1 | + 8·5-s + 2·13-s − 8·17-s + 38·25-s − 18·37-s − 5·49-s − 16·53-s − 10·61-s + 16·65-s + 2·73-s − 64·85-s + 24·89-s + 10·97-s − 28·109-s + 24·113-s − 6·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 0.554·13-s − 1.94·17-s + 38/5·25-s − 2.95·37-s − 5/7·49-s − 2.19·53-s − 1.28·61-s + 1.98·65-s + 0.234·73-s − 6.94·85-s + 2.54·89-s + 1.01·97-s − 2.68·109-s + 2.25·113-s − 0.545·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.781173969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781173969\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−9.397687695952564808383582699781, −9.353481303159343167683292410657, −8.806587663978251937370981617555, −8.539782519934566925988068508340, −7.50312413977084454112831481379, −6.65045650092432229207532369718, −6.44248673632066345518308551587, −6.23424820798207238945321961084, −5.45381403020204823365032144634, −5.10755730919050596352378085939, −4.60013470795046618235360781298, −3.36478379045069074059258347613, −2.61810189422413157391267840739, −1.81832624891237728767312244270, −1.69047687282002906512933211555,
1.69047687282002906512933211555, 1.81832624891237728767312244270, 2.61810189422413157391267840739, 3.36478379045069074059258347613, 4.60013470795046618235360781298, 5.10755730919050596352378085939, 5.45381403020204823365032144634, 6.23424820798207238945321961084, 6.44248673632066345518308551587, 6.65045650092432229207532369718, 7.50312413977084454112831481379, 8.539782519934566925988068508340, 8.806587663978251937370981617555, 9.353481303159343167683292410657, 9.397687695952564808383582699781
