| L(s) = 1 | + 2·5-s − 3·9-s + 8·19-s + 8·23-s + 3·25-s − 4·29-s − 16·43-s − 6·45-s + 8·47-s + 2·49-s + 12·53-s + 16·67-s − 12·73-s + 9·81-s + 16·95-s − 28·97-s + 12·101-s + 16·115-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 − 9-s + 1.83·19-s + 1.66·23-s + 3/5·25-s − 0.742·29-s − 2.43·43-s − 0.894·45-s + 1.16·47-s + 2/7·49-s + 1.64·53-s + 1.95·67-s − 1.40·73-s + 81-s + 1.64·95-s − 2.84·97-s + 1.19·101-s + 1.49·115-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.799134205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799134205\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.460240254627452220255642509658, −9.022035638415746178025932660186, −8.610742879040566128162935487774, −8.094484908266087825781152713303, −7.38986169737030489086018454093, −6.97238544391138654785050907556, −6.52411082007979697711723996792, −5.70093888866808737514223874840, −5.34437678528517976086556923571, −5.16731330798328126955314301729, −4.16841981012504174266678012358, −3.26180587408034592610487247010, −2.94471466186386712232384138589, −2.05937366765957406711596802667, −1.01814837175580913037691321841,
1.01814837175580913037691321841, 2.05937366765957406711596802667, 2.94471466186386712232384138589, 3.26180587408034592610487247010, 4.16841981012504174266678012358, 5.16731330798328126955314301729, 5.34437678528517976086556923571, 5.70093888866808737514223874840, 6.52411082007979697711723996792, 6.97238544391138654785050907556, 7.38986169737030489086018454093, 8.094484908266087825781152713303, 8.610742879040566128162935487774, 9.022035638415746178025932660186, 9.460240254627452220255642509658
