| L(s) = 1 | + 2·7-s + 2·9-s − 2·17-s − 3·23-s − 25-s + 12·31-s + 11·41-s + 19·47-s − 10·49-s + 4·63-s − 9·71-s − 15·73-s + 7·79-s − 5·81-s + 17·89-s − 3·97-s + 5·103-s − 23·113-s − 4·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2/3·9-s − 0.485·17-s − 0.625·23-s − 1/5·25-s + 2.15·31-s + 1.71·41-s + 2.77·47-s − 1.42·49-s + 0.503·63-s − 1.06·71-s − 1.75·73-s + 0.787·79-s − 5/9·81-s + 1.80·89-s − 0.304·97-s + 0.492·103-s − 2.16·113-s − 0.366·119-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.662492752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.662492752\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 281 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.805941476976490229958433110753, −9.360686394672445814357972516924, −8.763177392123102410273207105600, −8.323564009413121020812633310052, −7.70692096781297826811476904845, −7.43926194277188759717400090267, −6.73414326298409625734681581914, −6.12104222894730258370971768139, −5.71646159677920657817216116282, −4.78896312657338825553475834120, −4.44387523985484038002325721925, −3.92793859968606297549325354905, −2.86409134783481408558698657053, −2.16160601243205273803368331693, −1.12609264040592658042000790844,
1.12609264040592658042000790844, 2.16160601243205273803368331693, 2.86409134783481408558698657053, 3.92793859968606297549325354905, 4.44387523985484038002325721925, 4.78896312657338825553475834120, 5.71646159677920657817216116282, 6.12104222894730258370971768139, 6.73414326298409625734681581914, 7.43926194277188759717400090267, 7.70692096781297826811476904845, 8.323564009413121020812633310052, 8.763177392123102410273207105600, 9.360686394672445814357972516924, 9.805941476976490229958433110753
