| L(s) = 1 | − 2-s + 2·4-s − 5·8-s + 3·9-s − 4·11-s + 5·16-s − 3·18-s + 4·22-s − 8·23-s + 5·25-s + 4·29-s − 10·32-s + 6·36-s + 6·37-s − 24·43-s − 8·44-s + 8·46-s − 5·50-s + 10·53-s − 4·58-s + 17·64-s − 4·67-s + 32·71-s − 15·72-s − 6·74-s − 8·79-s + 24·86-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 1.76·8-s + 9-s − 1.20·11-s + 5/4·16-s − 0.707·18-s + 0.852·22-s − 1.66·23-s + 25-s + 0.742·29-s − 1.76·32-s + 36-s + 0.986·37-s − 3.65·43-s − 1.20·44-s + 1.17·46-s − 0.707·50-s + 1.37·53-s − 0.525·58-s + 17/8·64-s − 0.488·67-s + 3.79·71-s − 1.76·72-s − 0.697·74-s − 0.900·79-s + 2.58·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5339563072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5339563072\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−15.85212576093522570832154229519, −15.49241818306105495547348293749, −15.08039720005677721664988852131, −14.43344061342251990602688111731, −13.54999095046078507734294954532, −12.97664701560853149160404088547, −12.35110043118596423755352575705, −11.89293230051654397708049852819, −11.22716252147463454174169575444, −10.49157349351209535688193050598, −9.963629133540794207882785141408, −9.576821537056004788205838341040, −8.359819327199796891744287885900, −8.255622698155190429547567793677, −7.16251823162741675847358746285, −6.65752375968195805720507732929, −5.83768404749240717606545266475, −4.85098688399215011214238702043, −3.42987312905157534010236914006, −2.26623532713692530088546710173,
2.26623532713692530088546710173, 3.42987312905157534010236914006, 4.85098688399215011214238702043, 5.83768404749240717606545266475, 6.65752375968195805720507732929, 7.16251823162741675847358746285, 8.255622698155190429547567793677, 8.359819327199796891744287885900, 9.576821537056004788205838341040, 9.963629133540794207882785141408, 10.49157349351209535688193050598, 11.22716252147463454174169575444, 11.89293230051654397708049852819, 12.35110043118596423755352575705, 12.97664701560853149160404088547, 13.54999095046078507734294954532, 14.43344061342251990602688111731, 15.08039720005677721664988852131, 15.49241818306105495547348293749, 15.85212576093522570832154229519
