| L(s) = 1 | + 2·2-s + 3·4-s − 7-s + 4·8-s + 6·11-s − 2·14-s + 5·16-s + 12·22-s + 12·23-s − 25-s − 3·28-s − 12·29-s + 6·32-s + 4·37-s − 20·43-s + 18·44-s + 24·46-s − 6·49-s − 2·50-s − 18·53-s − 4·56-s − 24·58-s + 7·64-s + 28·67-s + 8·74-s − 6·77-s + 16·79-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s + 1.80·11-s − 0.534·14-s + 5/4·16-s + 2.55·22-s + 2.50·23-s − 1/5·25-s − 0.566·28-s − 2.22·29-s + 1.06·32-s + 0.657·37-s − 3.04·43-s + 2.71·44-s + 3.53·46-s − 6/7·49-s − 0.282·50-s − 2.47·53-s − 0.534·56-s − 3.15·58-s + 7/8·64-s + 3.42·67-s + 0.929·74-s − 0.683·77-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.259761832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.259761832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.507801646196580209679416383095, −9.022301181159976542679631550139, −8.153752740732516759811873175243, −7.82723843094092867969941089213, −6.90390849159694968636551446179, −6.71508725988629390136097161445, −6.51972259556560194206176883369, −5.69537931970611061608929469001, −5.14293667469015758909535382312, −4.77709333112371596512170551343, −4.01126047180402972127592124525, −3.38549178369413487864105189864, −3.26671376711039705407751493003, −2.07276435338156744028852711093, −1.33346620954041525748246641205,
1.33346620954041525748246641205, 2.07276435338156744028852711093, 3.26671376711039705407751493003, 3.38549178369413487864105189864, 4.01126047180402972127592124525, 4.77709333112371596512170551343, 5.14293667469015758909535382312, 5.69537931970611061608929469001, 6.51972259556560194206176883369, 6.71508725988629390136097161445, 6.90390849159694968636551446179, 7.82723843094092867969941089213, 8.153752740732516759811873175243, 9.022301181159976542679631550139, 9.507801646196580209679416383095
