| L(s) = 1 | − 3-s − 2·4-s − 2·9-s + 11-s + 2·12-s − 13-s + 4·16-s + 7·23-s − 6·25-s + 5·27-s − 33-s + 4·36-s − 13·37-s + 39-s − 2·44-s − 10·47-s − 4·48-s − 4·49-s + 2·52-s + 14·59-s − 2·61-s − 8·64-s − 7·69-s − 14·71-s + 5·73-s + 6·75-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 2/3·9-s + 0.301·11-s + 0.577·12-s − 0.277·13-s + 16-s + 1.45·23-s − 6/5·25-s + 0.962·27-s − 0.174·33-s + 2/3·36-s − 2.13·37-s + 0.160·39-s − 0.301·44-s − 1.45·47-s − 0.577·48-s − 4/7·49-s + 0.277·52-s + 1.82·59-s − 0.256·61-s − 64-s − 0.842·69-s − 1.66·71-s + 0.585·73-s + 0.692·75-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 409 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−9.730857832469110050088674933644, −9.147850183242265209824132216088, −8.787634512160940673297785685441, −8.252424729538542192709428217314, −7.82357660764592129393953375496, −6.87804980780317561326121511889, −6.72745087772409814473059032458, −5.70791144691087681719600667856, −5.44529213814396358265553611035, −4.91849311165026408936328059457, −4.22874534561839067227316089512, −3.50844519216733941950664943988, −2.83155880268884773480735287243, −1.47949848816749104257593862941, 0,
1.47949848816749104257593862941, 2.83155880268884773480735287243, 3.50844519216733941950664943988, 4.22874534561839067227316089512, 4.91849311165026408936328059457, 5.44529213814396358265553611035, 5.70791144691087681719600667856, 6.72745087772409814473059032458, 6.87804980780317561326121511889, 7.82357660764592129393953375496, 8.252424729538542192709428217314, 8.787634512160940673297785685441, 9.147850183242265209824132216088, 9.730857832469110050088674933644
