| L(s) = 1 | + 2-s − 2·3-s − 2·4-s − 2·5-s − 2·6-s + 2·7-s − 3·8-s + 3·9-s − 2·10-s + 4·12-s + 6·13-s + 2·14-s + 4·15-s + 16-s − 2·17-s + 3·18-s + 4·19-s + 4·20-s − 4·21-s + 4·23-s + 6·24-s + 3·25-s + 6·26-s − 4·27-s − 4·28-s + 2·29-s + 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 4-s − 0.894·5-s − 0.816·6-s + 0.755·7-s − 1.06·8-s + 9-s − 0.632·10-s + 1.15·12-s + 1.66·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.917·19-s + 0.894·20-s − 0.872·21-s + 0.834·23-s + 1.22·24-s + 3/5·25-s + 1.17·26-s − 0.769·27-s − 0.755·28-s + 0.371·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.893685617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.893685617\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 53 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 111 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 149 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 30 T + 414 T^{2} - 30 p T^{3} + p^{2} T^{4} \) |
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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
−9.325012064678932692986700752254, −9.037052855310290616352642500388, −8.582099751421481240773285062363, −8.460909499670961194537051937551, −7.79315117774579275858784669052, −7.47676152222131388133449886600, −7.13298187166834420041071977118, −6.50230302422962949453235552369, −6.04276253560562332359592948124, −5.87906373830787910810902376787, −5.19520931276398807688658271499, −5.00708051626643676653931728210, −4.48527348226146045173814460850, −4.30874286982828647610998319217, −3.70187628979622433491499798442, −3.48742788429213568835127407996, −2.74253557029201212677306955913, −1.83600078314410590847902660307, −0.830149055478123674746407958472, −0.78651131815844179878191431538,
0.78651131815844179878191431538, 0.830149055478123674746407958472, 1.83600078314410590847902660307, 2.74253557029201212677306955913, 3.48742788429213568835127407996, 3.70187628979622433491499798442, 4.30874286982828647610998319217, 4.48527348226146045173814460850, 5.00708051626643676653931728210, 5.19520931276398807688658271499, 5.87906373830787910810902376787, 6.04276253560562332359592948124, 6.50230302422962949453235552369, 7.13298187166834420041071977118, 7.47676152222131388133449886600, 7.79315117774579275858784669052, 8.460909499670961194537051937551, 8.582099751421481240773285062363, 9.037052855310290616352642500388, 9.325012064678932692986700752254
