| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s − 2·7-s + 2·10-s − 4·11-s − 2·14-s + 2·15-s − 2·21-s − 4·22-s + 25-s − 3·29-s + 2·30-s + 3·31-s − 32-s − 4·33-s − 4·35-s − 2·42-s + 49-s + 50-s − 3·53-s − 8·55-s − 3·58-s + 2·59-s + 3·62-s − 64-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 2·5-s + 6-s − 2·7-s + 2·10-s − 4·11-s − 2·14-s + 2·15-s − 2·21-s − 4·22-s + 25-s − 3·29-s + 2·30-s + 3·31-s − 32-s − 4·33-s − 4·35-s − 2·42-s + 49-s + 50-s − 3·53-s − 8·55-s − 3·58-s + 2·59-s + 3·62-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5821752625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5821752625\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 7 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 31 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 59 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−9.220460024291236800183252550501, −8.640116519774883854406429770228, −8.413873210885878341335181587206, −8.151981785243234070540589674822, −8.030554828472522973099657900585, −7.72003878295877714324438169378, −7.35641075404767738939443705765, −7.27846590546589113300465300916, −6.66737423905004273802908183672, −6.55079544240914588800912920405, −6.18417242379035783605378456177, −5.92494595969862360737800584681, −5.73815954725314739068205768804, −5.42176144190431520797850873741, −5.13230179327098417332151020187, −5.06943013704510882590567829381, −4.66102224057391094139786868486, −4.24771587458155608530068743414, −3.76032049143978445935635006511, −3.28960369502113406150239607143, −3.12520497354087074804149569494, −2.90109511654206965394973934551, −2.42222488969921082082691890645, −2.18476312149028855476357236627, −1.93107444314963751938538103688,
1.93107444314963751938538103688, 2.18476312149028855476357236627, 2.42222488969921082082691890645, 2.90109511654206965394973934551, 3.12520497354087074804149569494, 3.28960369502113406150239607143, 3.76032049143978445935635006511, 4.24771587458155608530068743414, 4.66102224057391094139786868486, 5.06943013704510882590567829381, 5.13230179327098417332151020187, 5.42176144190431520797850873741, 5.73815954725314739068205768804, 5.92494595969862360737800584681, 6.18417242379035783605378456177, 6.55079544240914588800912920405, 6.66737423905004273802908183672, 7.27846590546589113300465300916, 7.35641075404767738939443705765, 7.72003878295877714324438169378, 8.030554828472522973099657900585, 8.151981785243234070540589674822, 8.413873210885878341335181587206, 8.640116519774883854406429770228, 9.220460024291236800183252550501
