| L(s) = 1 | − 2-s + 4-s + 6·7-s − 8-s − 6·14-s + 16-s + 6·19-s + 8·25-s + 6·28-s − 2·29-s − 32-s − 6·38-s + 4·41-s + 14·49-s − 8·50-s + 4·53-s − 6·56-s + 2·58-s + 22·59-s + 6·61-s + 64-s + 14·71-s − 8·73-s + 6·76-s − 4·82-s − 18·89-s − 14·98-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 2.26·7-s − 0.353·8-s − 1.60·14-s + 1/4·16-s + 1.37·19-s + 8/5·25-s + 1.13·28-s − 0.371·29-s − 0.176·32-s − 0.973·38-s + 0.624·41-s + 2·49-s − 1.13·50-s + 0.549·53-s − 0.801·56-s + 0.262·58-s + 2.86·59-s + 0.768·61-s + 1/8·64-s + 1.66·71-s − 0.936·73-s + 0.688·76-s − 0.441·82-s − 1.90·89-s − 1.41·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.107044108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.107044108\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−8.480397071951097656995977938439, −8.204937487402916796471567126127, −7.72815597649003185894010701457, −7.30643948175919213397419248031, −6.94611961062574371317122745374, −6.40007833640321970929961970032, −5.52976853194995048770792407391, −5.22824871432729008196169201854, −5.00849320072731861488643482828, −4.19413029953658278788964422271, −3.74275101105643100013456370998, −2.77850342901391922975757706147, −2.32434390089717673535317243558, −1.41909076861570765389036760206, −1.02630107427529193309570785201,
1.02630107427529193309570785201, 1.41909076861570765389036760206, 2.32434390089717673535317243558, 2.77850342901391922975757706147, 3.74275101105643100013456370998, 4.19413029953658278788964422271, 5.00849320072731861488643482828, 5.22824871432729008196169201854, 5.52976853194995048770792407391, 6.40007833640321970929961970032, 6.94611961062574371317122745374, 7.30643948175919213397419248031, 7.72815597649003185894010701457, 8.204937487402916796471567126127, 8.480397071951097656995977938439
