| L(s) = 1 | + 8·7-s − 8·23-s − 8·25-s − 16·31-s − 16·41-s − 16·47-s + 34·49-s − 24·71-s − 4·73-s − 28·89-s − 32·97-s − 8·103-s + 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 64·161-s + 163-s + 167-s − 8·169-s + 173-s − 64·175-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1.66·23-s − 8/5·25-s − 2.87·31-s − 2.49·41-s − 2.33·47-s + 34/7·49-s − 2.84·71-s − 0.468·73-s − 2.96·89-s − 3.24·97-s − 0.788·103-s + 0.376·113-s − 1.27·121-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 − 5.04·161-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s − 4.83·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−7.53413814462915810572819498920, −7.45614250003428663403538213435, −6.83462245430182591544481842943, −6.75011339841494960825337771469, −5.94791332206254764964182323358, −5.75968663101382573921778867059, −5.34910991200909398195537998534, −5.25678139234700986992934751580, −4.82821336191601818257543150790, −4.33692588222372219242358122420, −4.03975531900275793108932488479, −3.94949651072399384362782792622, −3.23535578129751299135920327957, −2.84159011125453390325761343219, −2.08172054214496432443369192771, −1.77467576648189249407772828247, −1.54489307620531675471409139468, −1.48823445499472879462712629739, 0, 0,
1.48823445499472879462712629739, 1.54489307620531675471409139468, 1.77467576648189249407772828247, 2.08172054214496432443369192771, 2.84159011125453390325761343219, 3.23535578129751299135920327957, 3.94949651072399384362782792622, 4.03975531900275793108932488479, 4.33692588222372219242358122420, 4.82821336191601818257543150790, 5.25678139234700986992934751580, 5.34910991200909398195537998534, 5.75968663101382573921778867059, 5.94791332206254764964182323358, 6.75011339841494960825337771469, 6.83462245430182591544481842943, 7.45614250003428663403538213435, 7.53413814462915810572819498920
