| L(s) = 1 | − 4·5-s + 9-s + 8·11-s − 12·19-s + 5·25-s − 24·29-s − 4·31-s − 32·41-s − 4·45-s − 32·55-s − 8·59-s + 28·61-s − 16·89-s + 48·95-s + 8·99-s + 32·101-s − 28·109-s + 38·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 96·145-s + 149-s + 151-s + 16·155-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s + 2.41·11-s − 2.75·19-s + 25-s − 4.45·29-s − 0.718·31-s − 4.99·41-s − 0.596·45-s − 4.31·55-s − 1.04·59-s + 3.58·61-s − 1.69·89-s + 4.92·95-s + 0.804·99-s + 3.18·101-s − 2.68·109-s + 3.45·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 7.97·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2513659431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2513659431\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 78 T^{2} + 3875 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.28184233568045228983144273209, −5.98596763636294739369667124980, −5.94484160721340019806129844978, −5.48459609624045875983809038329, −5.42494547525023399183097022369, −5.15339370950082611779571595839, −5.00645988828839004505332497590, −4.53704001163914126069939408627, −4.52195087385897837808316065985, −4.30620370740051592516843906661, −3.91217647509236235378160640900, −3.83224199457062723471694908881, −3.69454764984263134861524948730, −3.62763645680479679820125314504, −3.59317916050644098694125908520, −3.03023538235333097740385598135, −2.93797129082118031697706793239, −2.23241214561685318696293250650, −2.14241762726067597876128037637, −1.83928915122763683636838296812, −1.69542043482815639547967671937, −1.57630842340526330756933027956, −1.04572725777777401426611315205, −0.42934143909950737326126646051, −0.12894544989900455822454417214,
0.12894544989900455822454417214, 0.42934143909950737326126646051, 1.04572725777777401426611315205, 1.57630842340526330756933027956, 1.69542043482815639547967671937, 1.83928915122763683636838296812, 2.14241762726067597876128037637, 2.23241214561685318696293250650, 2.93797129082118031697706793239, 3.03023538235333097740385598135, 3.59317916050644098694125908520, 3.62763645680479679820125314504, 3.69454764984263134861524948730, 3.83224199457062723471694908881, 3.91217647509236235378160640900, 4.30620370740051592516843906661, 4.52195087385897837808316065985, 4.53704001163914126069939408627, 5.00645988828839004505332497590, 5.15339370950082611779571595839, 5.42494547525023399183097022369, 5.48459609624045875983809038329, 5.94484160721340019806129844978, 5.98596763636294739369667124980, 6.28184233568045228983144273209
