| L(s) = 1 | + 2·2-s + 10·3-s − 5·5-s + 20·6-s − 8·8-s + 27·9-s − 10·10-s − 53·11-s − 50·13-s − 50·15-s − 16·16-s + 14·17-s + 54·18-s − 95·19-s − 106·22-s − 23-s − 80·24-s − 100·26-s − 190·27-s − 412·29-s − 100·30-s + 108·31-s − 530·33-s + 28·34-s + 57·37-s − 190·38-s − 500·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s − 0.447·5-s + 1.36·6-s − 0.353·8-s + 9-s − 0.316·10-s − 1.45·11-s − 1.06·13-s − 0.860·15-s − 1/4·16-s + 0.199·17-s + 0.707·18-s − 1.14·19-s − 1.02·22-s − 0.00906·23-s − 0.680·24-s − 0.754·26-s − 1.35·27-s − 2.63·29-s − 0.608·30-s + 0.625·31-s − 2.79·33-s + 0.141·34-s + 0.253·37-s − 0.811·38-s − 2.05·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.353009370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353009370\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 10 T + 73 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 53 T + 1478 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 25 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T - 4717 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 p T + 6 p^{2} T^{2} + 5 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 12166 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 206 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 108 T - 18127 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 57 T - 47404 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 243 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 434 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 231 T - 50462 T^{2} + 231 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 263 T - 79708 T^{2} + 263 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 24 T - 204803 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 116 T - 213525 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 204 T - 259147 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 484 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 692 T + 89847 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 466 T - 275883 T^{2} + 466 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 228 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 362 T - 573925 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 854 T + p^{3} T^{2} )^{2} \) |
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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
−10.77082902395692560891285584274, −10.23688378925989346792945135914, −9.711661750743829315605755925807, −9.394969239227515617716335388524, −8.886148322993568870120091960568, −8.505997016195774456057222541233, −7.999352762799497674328620029805, −7.77674634886274600532663943263, −7.35829880519985272699916093692, −6.81164520679210897722858169574, −5.88689582456059279781441253962, −5.58092264679079218858478342873, −5.01066332476458470956326969785, −4.32478136805557715532982590980, −3.91279655308931221229062052847, −3.27640923451254061856015313000, −2.85539040298633588255368351043, −2.32164983013291976961065628361, −1.92279758916243896096782582503, −0.25709458749117034338385768835,
0.25709458749117034338385768835, 1.92279758916243896096782582503, 2.32164983013291976961065628361, 2.85539040298633588255368351043, 3.27640923451254061856015313000, 3.91279655308931221229062052847, 4.32478136805557715532982590980, 5.01066332476458470956326969785, 5.58092264679079218858478342873, 5.88689582456059279781441253962, 6.81164520679210897722858169574, 7.35829880519985272699916093692, 7.77674634886274600532663943263, 7.999352762799497674328620029805, 8.505997016195774456057222541233, 8.886148322993568870120091960568, 9.394969239227515617716335388524, 9.711661750743829315605755925807, 10.23688378925989346792945135914, 10.77082902395692560891285584274
