| L(s) = 1 | + 2-s + 4·4-s + 12·5-s + 11·8-s + 12·10-s + 10·11-s + 11·16-s − 36·19-s + 48·20-s + 10·22-s − 14·23-s + 71·25-s + 76·29-s − 48·31-s + 44·32-s − 26·37-s − 36·38-s + 132·40-s + 52·43-s + 40·44-s − 14·46-s − 48·47-s + 71·50-s + 10·53-s + 120·55-s + 76·58-s + 132·59-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 4-s + 12/5·5-s + 11/8·8-s + 6/5·10-s + 0.909·11-s + 0.687·16-s − 1.89·19-s + 12/5·20-s + 5/11·22-s − 0.608·23-s + 2.83·25-s + 2.62·29-s − 1.54·31-s + 11/8·32-s − 0.702·37-s − 0.947·38-s + 3.29·40-s + 1.20·43-s + 0.909·44-s − 0.304·46-s − 1.02·47-s + 1.41·50-s + 0.188·53-s + 2.18·55-s + 1.31·58-s + 2.23·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.590297361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.590297361\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - 3 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T - 21 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 36 T + 793 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T - 333 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 48 T + 1729 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 73 T + p^{2} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 1438 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 48 T + 2977 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T - 2709 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 132 T + 9289 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 60 T + 4921 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 74 T + 987 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 72 T + 7057 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 46 T - 4125 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 5666 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 72 T + 9649 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15746 T^{2} + p^{4} T^{4} \) |
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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
−10.83438409694830611493282757701, −10.69759836808215586022693512268, −10.34903231965055891065700284173, −9.912160965546718488244175166171, −9.383505416595600936977746882573, −9.085190660469784382831280214244, −8.257270005857351800051446366685, −8.151789428144310965399377104746, −7.06090669586576222517234086528, −6.67695552250857653920362859304, −6.48833883359041420261515689477, −6.11503531307071872488290226680, −5.30135834975005129190393480323, −5.19570578340958736209463501990, −4.22467728464878507853117327892, −3.89777289119454904851101034952, −2.68298520754467771256277597183, −2.28140234775992907826946839430, −1.83174956887200399018616048715, −1.17420514568204544171694956418,
1.17420514568204544171694956418, 1.83174956887200399018616048715, 2.28140234775992907826946839430, 2.68298520754467771256277597183, 3.89777289119454904851101034952, 4.22467728464878507853117327892, 5.19570578340958736209463501990, 5.30135834975005129190393480323, 6.11503531307071872488290226680, 6.48833883359041420261515689477, 6.67695552250857653920362859304, 7.06090669586576222517234086528, 8.151789428144310965399377104746, 8.257270005857351800051446366685, 9.085190660469784382831280214244, 9.383505416595600936977746882573, 9.912160965546718488244175166171, 10.34903231965055891065700284173, 10.69759836808215586022693512268, 10.83438409694830611493282757701
