| L(s) = 1 | + 2·2-s − 10·4-s − 20·7-s − 32·8-s + 22·11-s − 80·13-s − 40·14-s + 44·16-s − 124·17-s + 72·19-s + 44·22-s − 98·23-s − 160·26-s + 200·28-s − 144·29-s − 34·31-s + 248·32-s − 248·34-s − 54·37-s + 144·38-s − 536·41-s + 60·43-s − 220·44-s − 196·46-s − 272·47-s − 338·49-s + 800·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 5/4·4-s − 1.07·7-s − 1.41·8-s + 0.603·11-s − 1.70·13-s − 0.763·14-s + 0.687·16-s − 1.76·17-s + 0.869·19-s + 0.426·22-s − 0.888·23-s − 1.20·26-s + 1.34·28-s − 0.922·29-s − 0.196·31-s + 1.37·32-s − 1.25·34-s − 0.239·37-s + 0.614·38-s − 2.04·41-s + 0.212·43-s − 0.753·44-s − 0.628·46-s − 0.844·47-s − 0.985·49-s + 2.13·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09436412526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09436412526\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 7 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 738 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 13238 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 72 T + 4214 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 98 T + 22847 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 144 T + 44554 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 34 T + 57519 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 54 T + 101843 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 536 T + 209618 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 60 T + 159146 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 272 T + 182942 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 492 T + 348862 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 634 T + 458975 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 840 T + 528794 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 754 T + 742455 T^{2} + 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 678 T + 813415 T^{2} - 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 400 T + 160962 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 p T - 279966 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 468 T + 1155130 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1842 T + 1935427 T^{2} - 1842 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2194 T + 2966547 T^{2} + 2194 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.845453733810344542760511320816, −8.546297645305899708440387882540, −7.84847147449465822319118412791, −7.81678909020413053207576090329, −7.08923631205891256304624426717, −6.76312471948227957871965258725, −6.36484366068527356967868924771, −6.17841101784755848011645259950, −5.34067623115730663760787883670, −5.25156796246307937555808921213, −4.71680353943433881963542093996, −4.57765227154553697146636426464, −3.85317271792736817146173714985, −3.70536313019727248385415343602, −3.15533778087021766259387709646, −2.80463247367345617896671556252, −1.98349637097142033597552798202, −1.71834942272120858071101274549, −0.59504197076450758566611580631, −0.085896833210309871970054469928,
0.085896833210309871970054469928, 0.59504197076450758566611580631, 1.71834942272120858071101274549, 1.98349637097142033597552798202, 2.80463247367345617896671556252, 3.15533778087021766259387709646, 3.70536313019727248385415343602, 3.85317271792736817146173714985, 4.57765227154553697146636426464, 4.71680353943433881963542093996, 5.25156796246307937555808921213, 5.34067623115730663760787883670, 6.17841101784755848011645259950, 6.36484366068527356967868924771, 6.76312471948227957871965258725, 7.08923631205891256304624426717, 7.81678909020413053207576090329, 7.84847147449465822319118412791, 8.546297645305899708440387882540, 8.845453733810344542760511320816
