| L(s) = 1 | + 2·2-s + 2·4-s − 16·7-s + 8·11-s + 6·13-s − 32·14-s − 4·16-s − 38·17-s + 16·22-s + 40·23-s + 12·26-s − 32·28-s − 88·31-s − 8·32-s − 76·34-s + 6·37-s + 140·41-s − 72·43-s + 16·44-s + 80·46-s + 128·49-s + 12·52-s − 34·53-s + 144·61-s − 176·62-s − 8·64-s − 88·67-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s − 2.28·7-s + 8/11·11-s + 6/13·13-s − 2.28·14-s − 1/4·16-s − 2.23·17-s + 8/11·22-s + 1.73·23-s + 6/13·26-s − 8/7·28-s − 2.83·31-s − 1/4·32-s − 2.23·34-s + 6/37·37-s + 3.41·41-s − 1.67·43-s + 4/11·44-s + 1.73·46-s + 2.61·49-s + 3/13·52-s − 0.641·53-s + 2.36·61-s − 2.83·62-s − 1/8·64-s − 1.31·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.685882721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685882721\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 38 T + 722 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 658 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 40 T + 800 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 238 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 72 T + 2592 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 1502 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 88 T + 3872 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 88 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 110 T + 6050 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12338 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15166 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 114 T + 6498 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.16735063093424290702195738466, −10.79785339661914097149225309578, −10.29698093800720513732218259981, −9.665430619152321039611415685836, −9.120781526493720330849209139236, −8.951584402564481257549442833166, −8.858544468615040273682033763957, −7.61179383011849509141596673671, −7.12654776820321573802488581568, −6.85815010741146095588493947741, −6.20086433134799526670751983998, −6.17550557877759553897974421321, −5.45469971713568904307878173902, −4.78053673542332195963079660501, −4.01692006462026777210756038754, −3.88639314077383374462759074766, −3.06481587553087848869698363139, −2.75910930754673675747905998697, −1.78827824898256082582368843122, −0.44305221038273263766191932166,
0.44305221038273263766191932166, 1.78827824898256082582368843122, 2.75910930754673675747905998697, 3.06481587553087848869698363139, 3.88639314077383374462759074766, 4.01692006462026777210756038754, 4.78053673542332195963079660501, 5.45469971713568904307878173902, 6.17550557877759553897974421321, 6.20086433134799526670751983998, 6.85815010741146095588493947741, 7.12654776820321573802488581568, 7.61179383011849509141596673671, 8.858544468615040273682033763957, 8.951584402564481257549442833166, 9.120781526493720330849209139236, 9.665430619152321039611415685836, 10.29698093800720513732218259981, 10.79785339661914097149225309578, 11.16735063093424290702195738466
