| L(s) = 1 | + 4-s − 2·7-s − 14·13-s + 4·16-s − 4·19-s − 14·25-s − 2·28-s − 16·31-s − 10·37-s + 20·43-s + 49-s − 14·52-s − 22·61-s + 11·64-s + 8·67-s + 44·73-s − 4·76-s − 40·79-s + 28·91-s + 20·97-s − 14·100-s + 56·103-s + 56·109-s − 8·112-s + 10·121-s − 16·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 3.88·13-s + 16-s − 0.917·19-s − 2.79·25-s − 0.377·28-s − 2.87·31-s − 1.64·37-s + 3.04·43-s + 1/7·49-s − 1.94·52-s − 2.81·61-s + 11/8·64-s + 0.977·67-s + 5.14·73-s − 0.458·76-s − 4.50·79-s + 2.93·91-s + 2.03·97-s − 7/5·100-s + 5.51·103-s + 5.36·109-s − 0.755·112-s + 0.909·121-s − 1.43·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7611250418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7611250418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 31 T^{2} + 672 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 2 T^{2} - 525 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 31 T^{2} + 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 34 T^{2} - 3885 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 14 T^{2} - 7725 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.36790168876185996470632654539, −7.31510403796565934090400314939, −7.26393452904838335317788472703, −6.54659150108289814211100742237, −6.38096670415992187932481796385, −6.26543208127739050967161965796, −5.99363739400003548678843416378, −5.68294121366445740896692083279, −5.50859722804966558107434022077, −5.29836449809563905216953318713, −4.97011593074884967125650999669, −4.84043267474088299825163429718, −4.47265051291275435560780946726, −4.23152097131733239029232234279, −3.92319260887807970641788507027, −3.59484039541357220439622750070, −3.32654406421422988550675445233, −3.27251300262659151054479943105, −2.63586236714681663275300973439, −2.42861316945270179595952779156, −2.02791339518687078126771134520, −1.93501957357203957978429151002, −1.87535001926669997039883383265, −0.67858585012376630527626015745, −0.28616830508994081669467467741,
0.28616830508994081669467467741, 0.67858585012376630527626015745, 1.87535001926669997039883383265, 1.93501957357203957978429151002, 2.02791339518687078126771134520, 2.42861316945270179595952779156, 2.63586236714681663275300973439, 3.27251300262659151054479943105, 3.32654406421422988550675445233, 3.59484039541357220439622750070, 3.92319260887807970641788507027, 4.23152097131733239029232234279, 4.47265051291275435560780946726, 4.84043267474088299825163429718, 4.97011593074884967125650999669, 5.29836449809563905216953318713, 5.50859722804966558107434022077, 5.68294121366445740896692083279, 5.99363739400003548678843416378, 6.26543208127739050967161965796, 6.38096670415992187932481796385, 6.54659150108289814211100742237, 7.26393452904838335317788472703, 7.31510403796565934090400314939, 7.36790168876185996470632654539
