L(s) = 1 | − 6·3-s − 2·4-s + 6·5-s − 2·7-s + 21·9-s − 6·11-s + 12·12-s − 36·15-s − 12·20-s + 12·21-s + 17·25-s − 54·27-s + 4·28-s − 36·29-s − 30·31-s + 36·33-s − 12·35-s − 42·36-s + 12·44-s + 126·45-s + 7·49-s − 6·53-s − 36·55-s + 18·59-s + 72·60-s − 42·63-s + 8·64-s + ⋯ |
L(s) = 1 | − 3.46·3-s − 4-s + 2.68·5-s − 0.755·7-s + 7·9-s − 1.80·11-s + 3.46·12-s − 9.29·15-s − 2.68·20-s + 2.61·21-s + 17/5·25-s − 10.3·27-s + 0.755·28-s − 6.68·29-s − 5.38·31-s + 6.26·33-s − 2.02·35-s − 7·36-s + 1.80·44-s + 18.7·45-s + 49-s − 0.824·53-s − 4.85·55-s + 2.34·59-s + 9.29·60-s − 5.29·63-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.04837167969\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04837167969\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 18 T + 137 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 - 10 T^{2} + p^{2} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 30 T + 383 T^{2} - 30 p T^{3} + p^{2} T^{4} )( 1 + 30 T + 383 T^{2} + 30 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
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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
−9.463437604897872984536497197881, −9.377064313900649788573689064210, −9.286755719502426704515534792916, −8.771094644198578161487309569184, −8.511580794288346771858191673026, −7.63798360644569760032079635865, −7.52877949431082005621226028509, −7.17665805796150911591748976546, −7.11434216307483274562168947597, −6.98257942465056893820267452275, −6.08847587932599529403997670380, −6.01063201128961134553312901332, −5.89745629144618398584096662281, −5.58776190811996915368453182123, −5.41072417612169412885880584733, −5.34860888934293696943476691805, −5.06325435010082705782695807136, −4.73995491413624673763265631232, −3.93336120093502218669534907081, −3.70829686570243310677140819631, −3.57254652042876278285779701688, −2.14401427075237738812491018014, −1.97590010121328969886745868285, −1.77186158140741299978715041944, −0.18704754528549063127061355071,
0.18704754528549063127061355071, 1.77186158140741299978715041944, 1.97590010121328969886745868285, 2.14401427075237738812491018014, 3.57254652042876278285779701688, 3.70829686570243310677140819631, 3.93336120093502218669534907081, 4.73995491413624673763265631232, 5.06325435010082705782695807136, 5.34860888934293696943476691805, 5.41072417612169412885880584733, 5.58776190811996915368453182123, 5.89745629144618398584096662281, 6.01063201128961134553312901332, 6.08847587932599529403997670380, 6.98257942465056893820267452275, 7.11434216307483274562168947597, 7.17665805796150911591748976546, 7.52877949431082005621226028509, 7.63798360644569760032079635865, 8.511580794288346771858191673026, 8.771094644198578161487309569184, 9.286755719502426704515534792916, 9.377064313900649788573689064210, 9.463437604897872984536497197881