| L(s) = 1 | − 2·2-s + 6·3-s + 2·4-s − 12·6-s − 6·7-s + 18·9-s + 24·11-s + 12·12-s − 24·13-s + 12·14-s − 4·16-s + 24·17-s − 36·18-s − 36·21-s − 48·22-s + 6·23-s + 48·26-s + 54·27-s − 12·28-s − 16·31-s + 8·32-s + 144·33-s − 48·34-s + 36·36-s − 96·37-s − 144·39-s − 96·41-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s + 1/2·4-s − 2·6-s − 6/7·7-s + 2·9-s + 2.18·11-s + 12-s − 1.84·13-s + 6/7·14-s − 1/4·16-s + 1.41·17-s − 2·18-s − 1.71·21-s − 2.18·22-s + 6/23·23-s + 1.84·26-s + 2·27-s − 3/7·28-s − 0.516·31-s + 1/4·32-s + 4.36·33-s − 1.41·34-s + 36-s − 2.59·37-s − 3.69·39-s − 2.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.319231412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319231412\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 96 T + 4608 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T + 1458 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3362 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10882 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 186 T + 17298 T^{2} - 186 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14942 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−15.39375256660699070522923994612, −14.82993393552372711007767059544, −14.64661998336004243774261294396, −13.92663883420789206712279376832, −13.75151866857549705537000314050, −12.69076772712617704935233109361, −12.04540692977967871954181129679, −11.86245207177202507846598702559, −10.29476229216911077361313726089, −10.08745848297075308675851995583, −9.411306383803152646954322466506, −8.987771474630974893119655693889, −8.597359020093749534253980411125, −7.77591223426661159754654466015, −7.02697502479373260278127599454, −6.69598080613287623425340706945, −5.08200756604778021156898137371, −3.67919922508282971486074977192, −3.15730387915536506833046377401, −1.80025676486837353071318586815,
1.80025676486837353071318586815, 3.15730387915536506833046377401, 3.67919922508282971486074977192, 5.08200756604778021156898137371, 6.69598080613287623425340706945, 7.02697502479373260278127599454, 7.77591223426661159754654466015, 8.597359020093749534253980411125, 8.987771474630974893119655693889, 9.411306383803152646954322466506, 10.08745848297075308675851995583, 10.29476229216911077361313726089, 11.86245207177202507846598702559, 12.04540692977967871954181129679, 12.69076772712617704935233109361, 13.75151866857549705537000314050, 13.92663883420789206712279376832, 14.64661998336004243774261294396, 14.82993393552372711007767059544, 15.39375256660699070522923994612
