| L(s) = 1 | + 2·5-s + 9-s − 2·13-s − 6·25-s + 4·29-s − 8·41-s + 2·45-s − 49-s − 8·53-s − 6·61-s − 4·65-s − 12·73-s + 81-s + 12·89-s − 22·101-s − 28·109-s − 16·113-s − 2·117-s + 18·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s − 0.554·13-s − 6/5·25-s + 0.742·29-s − 1.24·41-s + 0.298·45-s − 1/7·49-s − 1.09·53-s − 0.768·61-s − 0.496·65-s − 1.40·73-s + 1/9·81-s + 1.27·89-s − 2.18·101-s − 2.68·109-s − 1.50·113-s − 0.184·117-s + 1.63·121-s − 1.96·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.82275695381107071802204626647, −7.67603791470543012378535765594, −6.97574145695481199343401429757, −6.58180442024545210539760054086, −6.25728064317721281876315522062, −5.63999352288440352173630201982, −5.34548954706251993153663823137, −4.76671377148386800939273200650, −4.31169109708857469720093362891, −3.74160812574869642442235154201, −3.06188414818360415434749672739, −2.53935016258453322383779478598, −1.84577017055069480056545953959, −1.36355327458576798945759641366, 0,
1.36355327458576798945759641366, 1.84577017055069480056545953959, 2.53935016258453322383779478598, 3.06188414818360415434749672739, 3.74160812574869642442235154201, 4.31169109708857469720093362891, 4.76671377148386800939273200650, 5.34548954706251993153663823137, 5.63999352288440352173630201982, 6.25728064317721281876315522062, 6.58180442024545210539760054086, 6.97574145695481199343401429757, 7.67603791470543012378535765594, 7.82275695381107071802204626647
