| L(s) = 1 | − 2·5-s + 9-s − 4·17-s − 25-s − 4·29-s + 4·37-s − 12·41-s − 2·45-s + 6·49-s − 8·53-s + 20·73-s + 81-s + 8·85-s + 12·89-s + 4·97-s − 12·101-s + 24·109-s + 24·113-s + 18·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s − 0.970·17-s − 1/5·25-s − 0.742·29-s + 0.657·37-s − 1.87·41-s − 0.298·45-s + 6/7·49-s − 1.09·53-s + 2.34·73-s + 1/9·81-s + 0.867·85-s + 1.27·89-s + 0.406·97-s − 1.19·101-s + 2.29·109-s + 2.25·113-s + 1.63·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.130958635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130958635\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.691923627437867948422828950830, −8.048041230269784869322916446860, −7.63944652645222027662443914462, −7.30855276395543262285989299456, −6.72121982423731190583539115864, −6.39119668976990184034077063122, −5.79288831539909707586128194961, −5.17742464001872899041951495744, −4.64481540547215630214918849795, −4.28148366980766797793469532950, −3.56945526963872524639101493836, −3.32067326836995459867850945301, −2.30839104780070361418713900584, −1.78585882254057721941277789615, −0.56361626273624854285563909815,
0.56361626273624854285563909815, 1.78585882254057721941277789615, 2.30839104780070361418713900584, 3.32067326836995459867850945301, 3.56945526963872524639101493836, 4.28148366980766797793469532950, 4.64481540547215630214918849795, 5.17742464001872899041951495744, 5.79288831539909707586128194961, 6.39119668976990184034077063122, 6.72121982423731190583539115864, 7.30855276395543262285989299456, 7.63944652645222027662443914462, 8.048041230269784869322916446860, 8.691923627437867948422828950830
