| L(s) = 1 | + 4-s − 10·7-s − 6·13-s − 2·25-s − 10·28-s + 18·31-s − 8·37-s + 22·43-s + 61·49-s − 6·52-s + 48·61-s − 64-s + 14·67-s − 16·79-s + 60·91-s + 24·97-s − 2·100-s − 30·103-s + 16·109-s + 14·121-s + 18·124-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.77·7-s − 1.66·13-s − 2/5·25-s − 1.88·28-s + 3.23·31-s − 1.31·37-s + 3.35·43-s + 61/7·49-s − 0.832·52-s + 6.14·61-s − 1/8·64-s + 1.71·67-s − 1.80·79-s + 6.28·91-s + 2.43·97-s − 1/5·100-s − 2.95·103-s + 1.53·109-s + 1.27·121-s + 1.61·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.067271434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067271434\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 49 T^{2} + 1560 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 43 T^{2} - 1632 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 154 T^{2} + 16827 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−6.84349754291025128298870157174, −6.68057058448482890671993454285, −6.67447731204549851154565640790, −6.64954984758675550743020885651, −6.19787768422033970859441032396, −5.80143880805302056902254115420, −5.75860688016330003624815282353, −5.57564867812986301531900882769, −5.47326716760131007573977296515, −5.04345516014552287439721530379, −4.57222185740111262567021036026, −4.46711590680808137196645100373, −4.22272551254325117153989942672, −3.85896870589754521225085252293, −3.75841836771058362514566126829, −3.42770597324321728871097104756, −2.99669529227892468521443642487, −2.95951665091381060431140018916, −2.77489431743503757169074812847, −2.33818536093999725187696294033, −2.12709260732711027798047961142, −2.09648493921121790217228276497, −0.866466918777364631281491154705, −0.70937677937849112180812066906, −0.52232413925073647578044356674,
0.52232413925073647578044356674, 0.70937677937849112180812066906, 0.866466918777364631281491154705, 2.09648493921121790217228276497, 2.12709260732711027798047961142, 2.33818536093999725187696294033, 2.77489431743503757169074812847, 2.95951665091381060431140018916, 2.99669529227892468521443642487, 3.42770597324321728871097104756, 3.75841836771058362514566126829, 3.85896870589754521225085252293, 4.22272551254325117153989942672, 4.46711590680808137196645100373, 4.57222185740111262567021036026, 5.04345516014552287439721530379, 5.47326716760131007573977296515, 5.57564867812986301531900882769, 5.75860688016330003624815282353, 5.80143880805302056902254115420, 6.19787768422033970859441032396, 6.64954984758675550743020885651, 6.67447731204549851154565640790, 6.68057058448482890671993454285, 6.84349754291025128298870157174
