| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 9-s − 10-s − 3·11-s − 13-s + 14-s + 16-s + 18-s + 20-s + 3·22-s − 4·25-s + 26-s − 28-s + 5·31-s − 32-s − 35-s − 36-s − 40-s + 12·43-s − 3·44-s − 45-s − 4·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.223·20-s + 0.639·22-s − 4/5·25-s + 0.196·26-s − 0.188·28-s + 0.898·31-s − 0.176·32-s − 0.169·35-s − 1/6·36-s − 0.158·40-s + 1.82·43-s − 0.452·44-s − 0.149·45-s − 0.583·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
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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
−9.176335402607272347844946194259, −8.554196491871512853080098650750, −8.048277201511952220384952214723, −7.59779859996626051565796096833, −7.28460925837971731290474682565, −6.46693423722979436929037600573, −6.10137273521328927651831024858, −5.69067431362360231194043511275, −4.99463341398052799086290250030, −4.43618241102087715034958955134, −3.57351897884574780499748956329, −2.80933067396245401530758659241, −2.41739089341694620195798362641, −1.41377015591121286018042482251, 0,
1.41377015591121286018042482251, 2.41739089341694620195798362641, 2.80933067396245401530758659241, 3.57351897884574780499748956329, 4.43618241102087715034958955134, 4.99463341398052799086290250030, 5.69067431362360231194043511275, 6.10137273521328927651831024858, 6.46693423722979436929037600573, 7.28460925837971731290474682565, 7.59779859996626051565796096833, 8.048277201511952220384952214723, 8.554196491871512853080098650750, 9.176335402607272347844946194259
