| L(s) = 1 | + 2-s − 4-s + 3·7-s − 3·8-s − 9-s + 3·14-s − 16-s − 4·17-s − 18-s − 4·23-s − 6·25-s − 3·28-s + 4·31-s + 5·32-s − 4·34-s + 36-s − 8·41-s − 4·46-s + 12·47-s + 6·49-s − 6·50-s − 9·56-s + 4·62-s − 3·63-s + 7·64-s + 4·68-s + 3·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.13·7-s − 1.06·8-s − 1/3·9-s + 0.801·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.834·23-s − 6/5·25-s − 0.566·28-s + 0.718·31-s + 0.883·32-s − 0.685·34-s + 1/6·36-s − 1.24·41-s − 0.589·46-s + 1.75·47-s + 6/7·49-s − 0.848·50-s − 1.20·56-s + 0.508·62-s − 0.377·63-s + 7/8·64-s + 0.485·68-s + 0.353·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9678252925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9678252925\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−12.36021754353015496421482418180, −11.98566448029874371175619227026, −11.52475335651447367015294906093, −10.83089289495071548852543718100, −10.17157672444718663560350386527, −9.408398385496674260626968138199, −8.780510706819587469543522642198, −8.225807961009441439699575191924, −7.64732326813100286750821949387, −6.57789503000444893301550976642, −5.85944151284085004982177107902, −5.13436599806522948692536302643, −4.43740873400926585273602040722, −3.70641959270425262073508709657, −2.28562209272240743597100597091,
2.28562209272240743597100597091, 3.70641959270425262073508709657, 4.43740873400926585273602040722, 5.13436599806522948692536302643, 5.85944151284085004982177107902, 6.57789503000444893301550976642, 7.64732326813100286750821949387, 8.225807961009441439699575191924, 8.780510706819587469543522642198, 9.408398385496674260626968138199, 10.17157672444718663560350386527, 10.83089289495071548852543718100, 11.52475335651447367015294906093, 11.98566448029874371175619227026, 12.36021754353015496421482418180
