| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 2·5-s + 4·6-s − 4·8-s + 3·9-s − 4·10-s − 2·11-s − 4·12-s + 8·13-s − 4·15-s + 8·16-s + 10·17-s − 6·18-s + 2·19-s + 4·20-s + 4·22-s − 6·23-s + 8·24-s + 3·25-s − 16·26-s − 4·27-s + 2·29-s + 8·30-s + 6·31-s − 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s + 1.63·6-s − 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s − 1.15·12-s + 2.21·13-s − 1.03·15-s + 2·16-s + 2.42·17-s − 1.41·18-s + 0.458·19-s + 0.894·20-s + 0.852·22-s − 1.25·23-s + 1.63·24-s + 3/5·25-s − 3.13·26-s − 0.769·27-s + 0.371·29-s + 1.46·30-s + 1.07·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8723285772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8723285772\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 87 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.27036769809479083075884081211, −10.22929369549912394541517448835, −9.778186748190145099850094740111, −9.569701230789221547532165236524, −8.837614841835761129453251935585, −8.554808346617920577059426734340, −7.981246231671966534200358295995, −7.86131857149675983830184212919, −7.16396310660547287368429365034, −6.44317236645923943628425439163, −6.19014696105195164971803590497, −5.83271814970507657037252491399, −5.51770006645377899765420329901, −5.05242130557099098677091489291, −4.04531268071808069611300017411, −3.49643118841849119941728596584, −2.95071109530728965709677561584, −2.02833646153387438431752175331, −1.02739748126678866703037333369, −0.907314652004903767442835188152,
0.907314652004903767442835188152, 1.02739748126678866703037333369, 2.02833646153387438431752175331, 2.95071109530728965709677561584, 3.49643118841849119941728596584, 4.04531268071808069611300017411, 5.05242130557099098677091489291, 5.51770006645377899765420329901, 5.83271814970507657037252491399, 6.19014696105195164971803590497, 6.44317236645923943628425439163, 7.16396310660547287368429365034, 7.86131857149675983830184212919, 7.981246231671966534200358295995, 8.554808346617920577059426734340, 8.837614841835761129453251935585, 9.569701230789221547532165236524, 9.778186748190145099850094740111, 10.22929369549912394541517448835, 10.27036769809479083075884081211
