| L(s) = 1 | − 2·4-s − 10·7-s + 3·16-s + 24·19-s + 7·25-s + 20·28-s − 8·37-s + 16·43-s + 61·49-s − 4·64-s + 56·67-s − 42·73-s − 48·76-s + 44·79-s − 24·97-s − 14·100-s − 36·103-s − 32·109-s − 30·112-s − 22·121-s + 127-s + 131-s − 240·133-s + 137-s + 139-s + 16·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s − 3.77·7-s + 3/4·16-s + 5.50·19-s + 7/5·25-s + 3.77·28-s − 1.31·37-s + 2.43·43-s + 61/7·49-s − 1/2·64-s + 6.84·67-s − 4.91·73-s − 5.50·76-s + 4.95·79-s − 2.43·97-s − 7/5·100-s − 3.54·103-s − 3.06·109-s − 2.83·112-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 20.8·133-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261443352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261443352\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 23 T^{2} - 312 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 70 T^{2} + 3219 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 97 T^{2} + 6600 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 134 T^{2} + 11067 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.05794904483952272791193211280, −6.80996001795398050724283337650, −6.66701968615863156889854001739, −6.46961552477754556325695534326, −6.06801380903661751535780622912, −5.90792420448656033212758635368, −5.58299091608529065642164586534, −5.38542151630930705020418797158, −5.19604708337470404041773207701, −5.16049894023288694875246249067, −5.10763409244156535404898720691, −4.31193199757985699995381258471, −3.98199708061392832995914317775, −3.87981853174853643816444169507, −3.85184067384947370926943337712, −3.31095347315847509005594815213, −3.17990500058952823133106132510, −3.14513429687092525675014953306, −2.74267906647261803193101578632, −2.61465838675511528612564514236, −2.25780566444298190212407930663, −1.30265569086316676921000972987, −1.00103493056925685697696194661, −0.912304857533416237477968961567, −0.35455714758699262879196960302,
0.35455714758699262879196960302, 0.912304857533416237477968961567, 1.00103493056925685697696194661, 1.30265569086316676921000972987, 2.25780566444298190212407930663, 2.61465838675511528612564514236, 2.74267906647261803193101578632, 3.14513429687092525675014953306, 3.17990500058952823133106132510, 3.31095347315847509005594815213, 3.85184067384947370926943337712, 3.87981853174853643816444169507, 3.98199708061392832995914317775, 4.31193199757985699995381258471, 5.10763409244156535404898720691, 5.16049894023288694875246249067, 5.19604708337470404041773207701, 5.38542151630930705020418797158, 5.58299091608529065642164586534, 5.90792420448656033212758635368, 6.06801380903661751535780622912, 6.46961552477754556325695534326, 6.66701968615863156889854001739, 6.80996001795398050724283337650, 7.05794904483952272791193211280
