| L(s) = 1 | + 6·3-s + 8·7-s + 21·9-s + 10·13-s + 4·16-s − 14·19-s + 48·21-s + 54·27-s − 14·31-s + 22·37-s + 60·39-s + 24·48-s + 23·49-s − 84·57-s + 168·63-s + 10·67-s + 34·73-s + 108·81-s + 80·91-s − 84·93-s − 28·97-s − 38·109-s + 132·111-s + 32·112-s + 210·117-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 3.02·7-s + 7·9-s + 2.77·13-s + 16-s − 3.21·19-s + 10.4·21-s + 10.3·27-s − 2.51·31-s + 3.61·37-s + 9.60·39-s + 3.46·48-s + 23/7·49-s − 11.1·57-s + 21.1·63-s + 1.22·67-s + 3.97·73-s + 12·81-s + 8.38·91-s − 8.71·93-s − 2.84·97-s − 3.63·109-s + 12.5·111-s + 3.02·112-s + 19.4·117-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(30.44428967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(30.44428967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 73 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 - 22 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 74 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p T^{2} )^{2}( 1 + 167 T^{2} + p^{2} T^{4} ) \) |
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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
−7.57521813512348543887760948784, −6.90615997761795287444352177962, −6.76499403295368720976560821925, −6.71961744510472894203788798444, −6.43803947594569118013155398879, −5.98283489973279875015884412358, −5.81697281989349410831356201589, −5.59437877827847246052872478297, −5.31171186897401487458429192904, −4.80017184354271570003084961541, −4.73878987124321131382674875776, −4.35448043420384922393670775123, −4.33722521695960552316982550276, −3.86136885959875517271304803711, −3.69555765151037959364210425504, −3.64422048473657452406481746743, −3.62600757946305135886453604994, −2.85976269043776192049029265803, −2.45623508298647302971280871973, −2.42232223699052523320168744110, −2.25523319644310250178072513440, −1.62351836894839337549805214989, −1.60615906871848476247335009106, −1.38377050582098042592754427721, −0.970339375842221279251400606925,
0.970339375842221279251400606925, 1.38377050582098042592754427721, 1.60615906871848476247335009106, 1.62351836894839337549805214989, 2.25523319644310250178072513440, 2.42232223699052523320168744110, 2.45623508298647302971280871973, 2.85976269043776192049029265803, 3.62600757946305135886453604994, 3.64422048473657452406481746743, 3.69555765151037959364210425504, 3.86136885959875517271304803711, 4.33722521695960552316982550276, 4.35448043420384922393670775123, 4.73878987124321131382674875776, 4.80017184354271570003084961541, 5.31171186897401487458429192904, 5.59437877827847246052872478297, 5.81697281989349410831356201589, 5.98283489973279875015884412358, 6.43803947594569118013155398879, 6.71961744510472894203788798444, 6.76499403295368720976560821925, 6.90615997761795287444352177962, 7.57521813512348543887760948784
