| L(s) = 1 | + 2·5-s + 12·13-s − 2·17-s − 4·19-s − 4·23-s + 5·25-s + 20·29-s + 8·31-s − 6·37-s + 4·41-s − 8·43-s + 8·47-s − 10·53-s + 12·59-s + 2·61-s + 24·65-s − 12·67-s + 24·71-s + 14·73-s + 8·79-s − 24·83-s − 4·85-s − 2·89-s − 8·95-s + 20·97-s − 6·101-s + 2·109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 3.32·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 25-s + 3.71·29-s + 1.43·31-s − 0.986·37-s + 0.624·41-s − 1.21·43-s + 1.16·47-s − 1.37·53-s + 1.56·59-s + 0.256·61-s + 2.97·65-s − 1.46·67-s + 2.84·71-s + 1.63·73-s + 0.900·79-s − 2.63·83-s − 0.433·85-s − 0.211·89-s − 0.820·95-s + 2.03·97-s − 0.597·101-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.127702942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.127702942\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.502822641816571771634306320565, −8.424920741814522036659944694045, −8.218986338123309159730323495793, −8.022155646319705380616939473410, −6.87736678257361120134973727372, −6.83372796223109581336696846951, −6.36788753780212138931747754294, −6.34310986327447485074712883525, −5.88001750581901972782425508733, −5.50272507745461089850190269729, −4.83657720312791131093659864617, −4.64780565590878982475590182341, −3.94983628381681533987193806165, −3.90004202929853511830446563241, −3.07618351361672951769536253124, −2.91546801257104559277777596806, −2.18618027198153737951585901659, −1.75605530083727811500754861500, −1.00277133427536567828942054591, −0.872398733712217342574418147342,
0.872398733712217342574418147342, 1.00277133427536567828942054591, 1.75605530083727811500754861500, 2.18618027198153737951585901659, 2.91546801257104559277777596806, 3.07618351361672951769536253124, 3.90004202929853511830446563241, 3.94983628381681533987193806165, 4.64780565590878982475590182341, 4.83657720312791131093659864617, 5.50272507745461089850190269729, 5.88001750581901972782425508733, 6.34310986327447485074712883525, 6.36788753780212138931747754294, 6.83372796223109581336696846951, 6.87736678257361120134973727372, 8.022155646319705380616939473410, 8.218986338123309159730323495793, 8.424920741814522036659944694045, 8.502822641816571771634306320565
