| L(s) = 1 | + 7·4-s − 12·7-s − 30·13-s + 33·16-s − 28·19-s − 84·28-s − 50·31-s − 108·37-s + 30·43-s + 10·49-s − 210·52-s + 88·61-s + 119·64-s − 132·67-s − 196·76-s + 74·79-s + 360·91-s − 156·97-s + 60·103-s + 188·109-s − 396·112-s − 199·121-s − 350·124-s + 127-s + 131-s + 336·133-s + 137-s + ⋯ |
| L(s) = 1 | + 7/4·4-s − 1.71·7-s − 2.30·13-s + 2.06·16-s − 1.47·19-s − 3·28-s − 1.61·31-s − 2.91·37-s + 0.697·43-s + 0.204·49-s − 4.03·52-s + 1.44·61-s + 1.85·64-s − 1.97·67-s − 2.57·76-s + 0.936·79-s + 3.95·91-s − 1.60·97-s + 0.582·103-s + 1.72·109-s − 3.53·112-s − 1.64·121-s − 2.82·124-s + 0.00787·127-s + 0.00763·131-s + 2.52·133-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7526106781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7526106781\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 199 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 15 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 49 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1009 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1673 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2786 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2017 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5422 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6062 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 66 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 9758 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 34 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 78 T + p^{2} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.55515140368313587612379541630, −10.16549930177166151846806349357, −9.840153024083425191777787244924, −9.137698253363532171559508745193, −9.074627167808279742331094854736, −8.206008727985710856779758779101, −7.73929181725265194078616909121, −7.10375530851576350297508242568, −7.07768851903849882397276958327, −6.61546654207155719910236159334, −6.28941811345417939653345749782, −5.52279126786190242032071402290, −5.34591380139513647164301674401, −4.48471918875353524655383383189, −3.78671761132145353525312951710, −3.14107318082554678426469383307, −2.87824639605964136864567599223, −2.01839508915350532294348399526, −1.92184613997572285045502988612, −0.27228894215499111607283521576,
0.27228894215499111607283521576, 1.92184613997572285045502988612, 2.01839508915350532294348399526, 2.87824639605964136864567599223, 3.14107318082554678426469383307, 3.78671761132145353525312951710, 4.48471918875353524655383383189, 5.34591380139513647164301674401, 5.52279126786190242032071402290, 6.28941811345417939653345749782, 6.61546654207155719910236159334, 7.07768851903849882397276958327, 7.10375530851576350297508242568, 7.73929181725265194078616909121, 8.206008727985710856779758779101, 9.074627167808279742331094854736, 9.137698253363532171559508745193, 9.840153024083425191777787244924, 10.16549930177166151846806349357, 10.55515140368313587612379541630
