| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 3·9-s + 2·12-s + 14-s + 16-s + 3·18-s + 8·19-s + 2·21-s + 2·24-s − 6·25-s + 4·27-s + 28-s − 4·29-s + 32-s + 3·36-s − 20·37-s + 8·38-s + 2·42-s + 2·48-s + 49-s − 6·50-s + 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 9-s + 0.577·12-s + 0.267·14-s + 1/4·16-s + 0.707·18-s + 1.83·19-s + 0.436·21-s + 0.408·24-s − 6/5·25-s + 0.769·27-s + 0.188·28-s − 0.742·29-s + 0.176·32-s + 1/2·36-s − 3.28·37-s + 1.29·38-s + 0.308·42-s + 0.288·48-s + 1/7·49-s − 0.848·50-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.759820155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.759820155\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.567727627901464767318320243097, −9.086022258671443790589095587058, −8.579197535720292899748335278003, −7.963562405403294064353399542187, −7.68532191797102726979476428936, −6.95470194591461271128583103192, −6.89499765960031010222506198709, −5.69761771888975023011122238972, −5.47990216984766844713395370376, −4.82857955948335988380640999352, −4.04752481141371955267217260485, −3.54052974946578322805924483948, −3.08481257245522303063597583848, −2.16063348217202322480816158281, −1.50800619952171259085588630937,
1.50800619952171259085588630937, 2.16063348217202322480816158281, 3.08481257245522303063597583848, 3.54052974946578322805924483948, 4.04752481141371955267217260485, 4.82857955948335988380640999352, 5.47990216984766844713395370376, 5.69761771888975023011122238972, 6.89499765960031010222506198709, 6.95470194591461271128583103192, 7.68532191797102726979476428936, 7.963562405403294064353399542187, 8.579197535720292899748335278003, 9.086022258671443790589095587058, 9.567727627901464767318320243097
