L(s) = 1 | + 4-s − 4·5-s + 6·11-s − 10·19-s − 4·20-s + 5·25-s − 16·29-s + 4·31-s − 12·41-s + 6·44-s − 2·49-s − 24·55-s + 8·59-s − 12·61-s − 64-s + 24·71-s − 10·76-s + 28·79-s − 4·89-s + 40·95-s + 5·100-s + 4·109-s − 16·116-s + 31·121-s + 4·124-s + 4·125-s + 127-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 1.78·5-s + 1.80·11-s − 2.29·19-s − 0.894·20-s + 25-s − 2.97·29-s + 0.718·31-s − 1.87·41-s + 0.904·44-s − 2/7·49-s − 3.23·55-s + 1.04·59-s − 1.53·61-s − 1/8·64-s + 2.84·71-s − 1.14·76-s + 3.15·79-s − 0.423·89-s + 4.10·95-s + 1/2·100-s + 0.383·109-s − 1.48·116-s + 2.81·121-s + 0.359·124-s + 0.357·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9337608153\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9337608153\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
good | 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.65210421784187620369172150453, −7.54838586190778368070410461149, −7.14335427235484732948682767883, −6.72586798548455458689580158667, −6.68217340010272629734577366937, −6.60336762947680891052400652300, −6.47491304887185274567948435795, −5.95779143696496832458662232213, −5.82244423395202215946973716866, −5.49352358648509082854347229653, −5.10647833000173110235147874424, −4.89667814164780581756798580039, −4.63856223281481133676216065331, −4.20680724143827621599311178753, −4.08517677212029222670744695388, −3.74547029634021612883503352458, −3.71291619005900959220767870373, −3.53026787957819039327232698862, −3.02857884017755800567062446034, −2.68473838534280655324056951404, −2.05113003131218847613026516857, −1.84720851107884458435957873793, −1.80966177423670436869841592481, −0.929724241238901098148039763664, −0.32279773276218022558764646914,
0.32279773276218022558764646914, 0.929724241238901098148039763664, 1.80966177423670436869841592481, 1.84720851107884458435957873793, 2.05113003131218847613026516857, 2.68473838534280655324056951404, 3.02857884017755800567062446034, 3.53026787957819039327232698862, 3.71291619005900959220767870373, 3.74547029634021612883503352458, 4.08517677212029222670744695388, 4.20680724143827621599311178753, 4.63856223281481133676216065331, 4.89667814164780581756798580039, 5.10647833000173110235147874424, 5.49352358648509082854347229653, 5.82244423395202215946973716866, 5.95779143696496832458662232213, 6.47491304887185274567948435795, 6.60336762947680891052400652300, 6.68217340010272629734577366937, 6.72586798548455458689580158667, 7.14335427235484732948682767883, 7.54838586190778368070410461149, 7.65210421784187620369172150453