| L(s) = 1 | + 4-s − 10·7-s + 12·13-s − 20·25-s − 10·28-s + 6·31-s + 16·37-s − 8·43-s + 61·49-s + 12·52-s + 48·61-s − 64-s − 4·67-s + 18·73-s + 26·79-s − 120·91-s + 30·97-s − 20·100-s + 16·109-s + 26·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.77·7-s + 3.32·13-s − 4·25-s − 1.88·28-s + 1.07·31-s + 2.63·37-s − 1.21·43-s + 61/7·49-s + 1.66·52-s + 6.14·61-s − 1/8·64-s − 0.488·67-s + 2.10·73-s + 2.92·79-s − 12.5·91-s + 3.04·97-s − 2·100-s + 1.53·109-s + 2.36·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.862456791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.862456791\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^3$ | \( 1 - 23 T^{2} - 312 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 26 T^{2} - 1005 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 14 T^{2} - 2013 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 139 T^{2} + 12432 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.82323949825183464593154193507, −6.59179256408968710387520566621, −6.57314191007925896585376998668, −6.33619587398958952524540107676, −6.05927797048809356416510584722, −6.01649664818177420919140427332, −5.97881594613866622739553479565, −5.72972261297425682344576767089, −5.40497848118695277787030388804, −4.97598731954459548205680918099, −4.83892901607449161539398547357, −4.16509659605412761393741852829, −3.98099765188957114360851619991, −3.84264637602976889375794164533, −3.73218731241469667515493026067, −3.54824882580925946953713407112, −3.31040816713884910735546582019, −3.12435435230595439166457848764, −2.56204477088151213747331425264, −2.27720520111199977494721844703, −2.22782047255857120914711178020, −1.83854381754686040360242480343, −1.02943547149077283385947772627, −0.68846744758648258084676093827, −0.57799652806191971808712768089,
0.57799652806191971808712768089, 0.68846744758648258084676093827, 1.02943547149077283385947772627, 1.83854381754686040360242480343, 2.22782047255857120914711178020, 2.27720520111199977494721844703, 2.56204477088151213747331425264, 3.12435435230595439166457848764, 3.31040816713884910735546582019, 3.54824882580925946953713407112, 3.73218731241469667515493026067, 3.84264637602976889375794164533, 3.98099765188957114360851619991, 4.16509659605412761393741852829, 4.83892901607449161539398547357, 4.97598731954459548205680918099, 5.40497848118695277787030388804, 5.72972261297425682344576767089, 5.97881594613866622739553479565, 6.01649664818177420919140427332, 6.05927797048809356416510584722, 6.33619587398958952524540107676, 6.57314191007925896585376998668, 6.59179256408968710387520566621, 6.82323949825183464593154193507
