| L(s) = 1 | + 3-s + 3·4-s + 2·7-s + 9-s + 6·11-s + 3·12-s + 3·13-s + 4·16-s − 6·17-s − 16·19-s + 2·21-s − 6·23-s + 3·25-s − 4·27-s + 6·28-s + 6·29-s + 6·33-s + 3·36-s − 21·37-s + 3·39-s − 3·41-s + 9·43-s + 18·44-s + 12·47-s + 4·48-s − 11·49-s − 6·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3/2·4-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.866·12-s + 0.832·13-s + 16-s − 1.45·17-s − 3.67·19-s + 0.436·21-s − 1.25·23-s + 3/5·25-s − 0.769·27-s + 1.13·28-s + 1.11·29-s + 1.04·33-s + 1/2·36-s − 3.45·37-s + 0.480·39-s − 0.468·41-s + 1.37·43-s + 2.71·44-s + 1.75·47-s + 0.577·48-s − 1.57·49-s − 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213302162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213302162\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 3 | $D_4\times C_2$ | \( 1 - T + 5 T^{3} - 11 T^{4} + 5 p T^{5} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 3 T^{2} + p T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3 T - 14 T^{2} + 9 T^{3} + 243 T^{4} + 9 p T^{5} - 14 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 180 T^{3} + 815 T^{4} + 180 p T^{5} + 42 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 72 T^{3} - 201 T^{4} - 72 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 21 T + 242 T^{2} + 1995 T^{3} + 13095 T^{4} + 1995 p T^{5} + 242 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 3 T - 28 T^{2} - 135 T^{3} - 681 T^{4} - 135 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 9 T - 20 T^{2} - 135 T^{3} + 4869 T^{4} - 135 p T^{5} - 20 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 147 T^{2} - 1188 T^{3} + 9848 T^{4} - 1188 p T^{5} + 147 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 195 T^{2} + 15077 T^{4} - 195 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 15 T + 56 T^{2} - 765 T^{3} + 11805 T^{4} - 765 p T^{5} + 56 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 9 T + 140 T^{2} + 1017 T^{3} + 10695 T^{4} + 1017 p T^{5} + 140 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 702 T^{3} + 9500 T^{4} - 702 p T^{5} + 129 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 6 T + 98 T^{2} + 516 T^{3} + 2943 T^{4} + 516 p T^{5} + 98 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 123 T^{2} + 11129 T^{4} - 123 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 9 T - 70 T^{2} - 243 T^{3} + 7671 T^{4} - 243 p T^{5} - 70 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−9.661949654379639047152329741874, −9.366138639387676657438105478759, −9.271776332815445164384279556556, −8.748335841571777423532440828848, −8.633645604063395384958235858077, −8.348138340646334577048604004039, −8.181253758822368333675662724942, −8.072372302863685151139341917754, −7.32817845101825303146798537557, −6.95578801786494056291031451433, −6.76004552245904836493891099464, −6.71100518752009920646199592344, −6.55459702392328134851439267664, −5.97132419106702627698070887920, −5.82430200736800529204205489112, −5.40136213073871425948978556209, −4.66889946572139557084964998279, −4.37869610020082997862464804339, −4.18050515916089938017271313011, −3.80565565100161112963210320760, −3.48608451435446061165039098249, −2.69831499622498658702115863073, −2.23676317545735293855296385601, −1.81875955252244691451777067763, −1.70766875367617418415368463340,
1.70766875367617418415368463340, 1.81875955252244691451777067763, 2.23676317545735293855296385601, 2.69831499622498658702115863073, 3.48608451435446061165039098249, 3.80565565100161112963210320760, 4.18050515916089938017271313011, 4.37869610020082997862464804339, 4.66889946572139557084964998279, 5.40136213073871425948978556209, 5.82430200736800529204205489112, 5.97132419106702627698070887920, 6.55459702392328134851439267664, 6.71100518752009920646199592344, 6.76004552245904836493891099464, 6.95578801786494056291031451433, 7.32817845101825303146798537557, 8.072372302863685151139341917754, 8.181253758822368333675662724942, 8.348138340646334577048604004039, 8.633645604063395384958235858077, 8.748335841571777423532440828848, 9.271776332815445164384279556556, 9.366138639387676657438105478759, 9.661949654379639047152329741874
