| L(s) = 1 | − 3·3-s − 5-s + 3·9-s − 11-s − 4·13-s + 3·15-s + 3·17-s − 5·19-s − 3·23-s + 5·25-s − 12·29-s + 31-s + 3·33-s + 5·37-s + 12·39-s + 20·41-s + 8·43-s − 3·45-s − 47-s − 9·51-s + 9·53-s + 55-s + 15·57-s − 3·59-s + 3·61-s + 4·65-s + 11·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 9-s − 0.301·11-s − 1.10·13-s + 0.774·15-s + 0.727·17-s − 1.14·19-s − 0.625·23-s + 25-s − 2.22·29-s + 0.179·31-s + 0.522·33-s + 0.821·37-s + 1.92·39-s + 3.12·41-s + 1.21·43-s − 0.447·45-s − 0.145·47-s − 1.26·51-s + 1.23·53-s + 0.134·55-s + 1.98·57-s − 0.390·59-s + 0.384·61-s + 0.496·65-s + 1.34·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3084491322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3084491322\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−10.81287541000616260925266606203, −10.19654751851798094008329726799, −9.657421074763476241599338425945, −9.468793693356120918791439686905, −8.805699277936019474710013521056, −8.375059857497353581552264913768, −7.60070243381425170115784259814, −7.56516224235763748853997438415, −7.09248415526312232094920061073, −6.44678880370783215481637996288, −5.95065094172571818152470857173, −5.57789199133190892364132919730, −5.46109827093559425374759838335, −4.68275428627776046058713422853, −4.11074067107696475497037067902, −3.99753842342516431449554238152, −2.72116242473673604830463391839, −2.51554974559482468729068402250, −1.33238404356840450097045312711, −0.32878365276115475990596103057,
0.32878365276115475990596103057, 1.33238404356840450097045312711, 2.51554974559482468729068402250, 2.72116242473673604830463391839, 3.99753842342516431449554238152, 4.11074067107696475497037067902, 4.68275428627776046058713422853, 5.46109827093559425374759838335, 5.57789199133190892364132919730, 5.95065094172571818152470857173, 6.44678880370783215481637996288, 7.09248415526312232094920061073, 7.56516224235763748853997438415, 7.60070243381425170115784259814, 8.375059857497353581552264913768, 8.805699277936019474710013521056, 9.468793693356120918791439686905, 9.657421074763476241599338425945, 10.19654751851798094008329726799, 10.81287541000616260925266606203
