| L(s) = 1 | − 2·5-s + 12·13-s + 12·17-s + 3·25-s + 4·29-s − 4·37-s + 12·41-s − 14·49-s − 12·53-s + 28·61-s − 24·65-s − 12·73-s − 24·85-s − 20·89-s + 4·97-s − 12·101-s − 36·109-s + 12·113-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 3.32·13-s + 2.91·17-s + 3/5·25-s + 0.742·29-s − 0.657·37-s + 1.87·41-s − 2·49-s − 1.64·53-s + 3.58·61-s − 2.97·65-s − 1.40·73-s − 2.60·85-s − 2.11·89-s + 0.406·97-s − 1.19·101-s − 3.44·109-s + 1.12·113-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.042100027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042100027\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.663099716937980035576099714984, −8.364107116024692137216049760263, −8.062184279727733628805054904271, −7.76560916456884806442341251241, −6.97039739431553724911307537358, −6.58285993178476791834968599347, −5.86178029126505651814373374846, −5.74168392413099570963868387782, −5.06868408963538865423952877412, −4.19803603229033482796144552370, −3.76264729663280098898303229481, −3.38984665882479408513314051846, −2.89778258507562116298634418143, −1.36054514292875137606341288502, −1.08829539378793470967143951854,
1.08829539378793470967143951854, 1.36054514292875137606341288502, 2.89778258507562116298634418143, 3.38984665882479408513314051846, 3.76264729663280098898303229481, 4.19803603229033482796144552370, 5.06868408963538865423952877412, 5.74168392413099570963868387782, 5.86178029126505651814373374846, 6.58285993178476791834968599347, 6.97039739431553724911307537358, 7.76560916456884806442341251241, 8.062184279727733628805054904271, 8.364107116024692137216049760263, 8.663099716937980035576099714984
