| L(s) = 1 | + 8·7-s − 12·17-s + 16·23-s − 25-s + 4·31-s − 8·41-s − 8·47-s + 34·49-s + 16·71-s + 20·73-s + 4·79-s + 8·89-s + 4·97-s + 16·103-s − 12·113-s − 96·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 128·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 2.91·17-s + 3.33·23-s − 1/5·25-s + 0.718·31-s − 1.24·41-s − 1.16·47-s + 34/7·49-s + 1.89·71-s + 2.34·73-s + 0.450·79-s + 0.847·89-s + 0.406·97-s + 1.57·103-s − 1.12·113-s − 8.80·119-s + 1.63·121-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 + 10.0·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.520598270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.520598270\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.426399870694915575858040605109, −9.310947730409124670028312621223, −8.780415604981976480543141779098, −8.492151149258021163955651641183, −8.204857537058191226664587579576, −7.924754241869277928116563874943, −7.11800879679482113766894365021, −7.09406695924732801407530110912, −6.58128994156822431887525295888, −6.12493573303369605110013983508, −5.13694568009224066415412906788, −5.07220509134546046623381450703, −4.78382163623525890830966552556, −4.54280225434132161333546905522, −3.84526034665342165625259758546, −3.18984239772692991727937110869, −2.30377915374674672098699154198, −2.14868804965802472093928077029, −1.46509190498058399378515711227, −0.811234539580455315129251649474,
0.811234539580455315129251649474, 1.46509190498058399378515711227, 2.14868804965802472093928077029, 2.30377915374674672098699154198, 3.18984239772692991727937110869, 3.84526034665342165625259758546, 4.54280225434132161333546905522, 4.78382163623525890830966552556, 5.07220509134546046623381450703, 5.13694568009224066415412906788, 6.12493573303369605110013983508, 6.58128994156822431887525295888, 7.09406695924732801407530110912, 7.11800879679482113766894365021, 7.924754241869277928116563874943, 8.204857537058191226664587579576, 8.492151149258021163955651641183, 8.780415604981976480543141779098, 9.310947730409124670028312621223, 9.426399870694915575858040605109
