| L(s) = 1 | − 2·4-s − 9-s − 2·13-s + 3·16-s − 2·17-s − 4·25-s − 2·29-s + 2·36-s − 2·37-s + 2·41-s + 4·52-s + 2·61-s − 4·64-s + 4·68-s − 2·73-s + 81-s − 2·89-s − 2·97-s + 8·100-s − 2·109-s + 8·113-s + 4·116-s + 2·117-s − 121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9-s − 2·13-s + 3·16-s − 2·17-s − 4·25-s − 2·29-s + 2·36-s − 2·37-s + 2·41-s + 4·52-s + 2·61-s − 4·64-s + 4·68-s − 2·73-s + 81-s − 2·89-s − 2·97-s + 8·100-s − 2·109-s + 8·113-s + 4·116-s + 2·117-s − 121-s + 127-s + 131-s + 137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1941718555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1941718555\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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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
−6.13234264343387051733691414331, −5.78589952608206587564666818171, −5.74230085572199996504755303173, −5.66336607753320709288782514045, −5.38785929894020550555921312428, −5.26161375166413064905502165549, −5.11123936771592119183489251876, −4.79714441016903614037997862214, −4.45839658958472568534696101512, −4.31015622373403251912049788081, −4.28797025778055333338983187684, −4.08278202384591875999618649238, −3.92508099109368381308759054178, −3.65369288460554169556934239256, −3.21979818444840351160258814261, −3.21880676564819533748686443537, −2.98486135222564180099534865166, −2.64718436470699227170840401798, −2.11525661783365455683824574787, −2.02792009985347986408124029241, −2.02035688785779943015252161692, −1.80767061983917839512624700675, −1.19116476975732416251713516843, −0.48573078504062185215331963776, −0.31312699152461624148476796347,
0.31312699152461624148476796347, 0.48573078504062185215331963776, 1.19116476975732416251713516843, 1.80767061983917839512624700675, 2.02035688785779943015252161692, 2.02792009985347986408124029241, 2.11525661783365455683824574787, 2.64718436470699227170840401798, 2.98486135222564180099534865166, 3.21880676564819533748686443537, 3.21979818444840351160258814261, 3.65369288460554169556934239256, 3.92508099109368381308759054178, 4.08278202384591875999618649238, 4.28797025778055333338983187684, 4.31015622373403251912049788081, 4.45839658958472568534696101512, 4.79714441016903614037997862214, 5.11123936771592119183489251876, 5.26161375166413064905502165549, 5.38785929894020550555921312428, 5.66336607753320709288782514045, 5.74230085572199996504755303173, 5.78589952608206587564666818171, 6.13234264343387051733691414331
