| L(s) = 1 | + 2·2-s + 3·3-s + 3·4-s + 2·5-s + 6·6-s + 7-s + 4·8-s + 2·9-s + 4·10-s + 2·11-s + 9·12-s − 2·13-s + 2·14-s + 6·15-s + 5·16-s + 6·17-s + 4·18-s − 5·19-s + 6·20-s + 3·21-s + 4·22-s + 13·23-s + 12·24-s − 2·25-s − 4·26-s − 6·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.894·5-s + 2.44·6-s + 0.377·7-s + 1.41·8-s + 2/3·9-s + 1.26·10-s + 0.603·11-s + 2.59·12-s − 0.554·13-s + 0.534·14-s + 1.54·15-s + 5/4·16-s + 1.45·17-s + 0.942·18-s − 1.14·19-s + 1.34·20-s + 0.654·21-s + 0.852·22-s + 2.71·23-s + 2.44·24-s − 2/5·25-s − 0.784·26-s − 1.15·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800964 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.25144729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.25144729\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 61 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 13 T + 87 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 147 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 167 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 30 T + 398 T^{2} + 30 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 19 T + 273 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
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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
−9.634126331381542459571156696462, −9.493853679170923711707262100218, −9.016658602990005590277194655382, −8.644788163759113683635177991488, −8.037627792011846380130734080206, −7.83546913147491549375525514035, −7.41512039326949315457536960330, −6.96485536256370916756438461341, −6.40570834177842911019681542203, −6.03683944736667851634956157477, −5.43166164504026626785493084882, −5.38701349981501582034839191934, −4.47587201385227917246782452243, −4.35136797229209741308935534869, −3.61055195245793942703063492052, −3.19710301862805042362604852835, −2.65100265788047042217550931325, −2.58572637566430163821537755543, −1.74499223070494619937004334258, −1.26339083872242493314239281878,
1.26339083872242493314239281878, 1.74499223070494619937004334258, 2.58572637566430163821537755543, 2.65100265788047042217550931325, 3.19710301862805042362604852835, 3.61055195245793942703063492052, 4.35136797229209741308935534869, 4.47587201385227917246782452243, 5.38701349981501582034839191934, 5.43166164504026626785493084882, 6.03683944736667851634956157477, 6.40570834177842911019681542203, 6.96485536256370916756438461341, 7.41512039326949315457536960330, 7.83546913147491549375525514035, 8.037627792011846380130734080206, 8.644788163759113683635177991488, 9.016658602990005590277194655382, 9.493853679170923711707262100218, 9.634126331381542459571156696462
