| L(s) = 1 | − 3·5-s + 3·7-s − 3·9-s + 3·11-s − 5·13-s − 9·23-s + 25-s − 3·29-s + 9·31-s − 9·35-s − 4·37-s − 9·41-s + 9·43-s + 9·45-s − 3·47-s − 49-s − 9·55-s − 3·59-s − 61-s − 9·63-s + 15·65-s + 15·67-s + 24·71-s − 4·73-s + 9·77-s + 15·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.13·7-s − 9-s + 0.904·11-s − 1.38·13-s − 1.87·23-s + 1/5·25-s − 0.557·29-s + 1.61·31-s − 1.52·35-s − 0.657·37-s − 1.40·41-s + 1.37·43-s + 1.34·45-s − 0.437·47-s − 1/7·49-s − 1.21·55-s − 0.390·59-s − 0.128·61-s − 1.13·63-s + 1.86·65-s + 1.83·67-s + 2.84·71-s − 0.468·73-s + 1.02·77-s + 1.68·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9793086280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9793086280\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.05242509159075290161575453242, −10.61849591041226634927842687526, −10.10094412228770490708811384366, −9.447472471398254403865837540016, −9.364152863457945047263330864997, −8.484754992454991743751328371017, −8.159478430115781344256916246864, −7.952748910165349935676321261264, −7.65067692156559365177176818819, −6.88992913993431250234446206440, −6.52196270576756211609268904887, −5.93992809808174445966961616651, −5.19514060554776249485979679895, −4.92339147054828808244483384241, −4.30905149124184812142372682830, −3.79434331752425458044215031825, −3.36675424568723974781347826549, −2.36528936015397940407342549446, −1.91230205601250973957560956072, −0.55367377142151674232286080885,
0.55367377142151674232286080885, 1.91230205601250973957560956072, 2.36528936015397940407342549446, 3.36675424568723974781347826549, 3.79434331752425458044215031825, 4.30905149124184812142372682830, 4.92339147054828808244483384241, 5.19514060554776249485979679895, 5.93992809808174445966961616651, 6.52196270576756211609268904887, 6.88992913993431250234446206440, 7.65067692156559365177176818819, 7.952748910165349935676321261264, 8.159478430115781344256916246864, 8.484754992454991743751328371017, 9.364152863457945047263330864997, 9.447472471398254403865837540016, 10.10094412228770490708811384366, 10.61849591041226634927842687526, 11.05242509159075290161575453242
