| L(s) = 1 | − 4-s − 5·7-s − 9·13-s + 16-s + 3·19-s − 4·25-s + 5·28-s − 6·31-s + 2·37-s + 5·43-s + 18·49-s + 9·52-s + 24·61-s − 64-s − 5·67-s + 15·73-s − 3·76-s + 79-s + 45·91-s − 3·97-s + 4·100-s − 6·103-s − 13·109-s − 5·112-s − 2·121-s + 6·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.88·7-s − 2.49·13-s + 1/4·16-s + 0.688·19-s − 4/5·25-s + 0.944·28-s − 1.07·31-s + 0.328·37-s + 0.762·43-s + 18/7·49-s + 1.24·52-s + 3.07·61-s − 1/8·64-s − 0.610·67-s + 1.75·73-s − 0.344·76-s + 0.112·79-s + 4.71·91-s − 0.304·97-s + 2/5·100-s − 0.591·103-s − 1.24·109-s − 0.472·112-s − 0.181·121-s + 0.538·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.67918371034966317341898739640, −7.29608215640852014938738524515, −6.88380319454323658351359926684, −6.67163933586349326953919726298, −5.89558908458079674080632563459, −5.54543966907262155729566369981, −5.21427216072771122642742868592, −4.62435568397508960024539676080, −4.01565633579559040588858246396, −3.64974973457130813398801330943, −3.05497654528752751281555583361, −2.51435450629834078345326389878, −2.09888124746657522409382749046, −0.73128835159777595123380153748, 0,
0.73128835159777595123380153748, 2.09888124746657522409382749046, 2.51435450629834078345326389878, 3.05497654528752751281555583361, 3.64974973457130813398801330943, 4.01565633579559040588858246396, 4.62435568397508960024539676080, 5.21427216072771122642742868592, 5.54543966907262155729566369981, 5.89558908458079674080632563459, 6.67163933586349326953919726298, 6.88380319454323658351359926684, 7.29608215640852014938738524515, 7.67918371034966317341898739640
