| L(s) = 1 | − 4·2-s + 10·4-s + 2·5-s − 20·8-s − 8·10-s + 8·11-s + 35·16-s + 10·17-s + 20·20-s − 32·22-s + 8·23-s + 5·25-s − 20·31-s − 56·32-s − 40·34-s − 40·40-s − 4·41-s + 2·43-s + 80·44-s − 32·46-s + 26·49-s − 20·50-s − 20·53-s + 16·55-s + 12·59-s + 8·61-s + 80·62-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s + 0.894·5-s − 7.07·8-s − 2.52·10-s + 2.41·11-s + 35/4·16-s + 2.42·17-s + 4.47·20-s − 6.82·22-s + 1.66·23-s + 25-s − 3.59·31-s − 9.89·32-s − 6.85·34-s − 6.32·40-s − 0.624·41-s + 0.304·43-s + 12.0·44-s − 4.71·46-s + 26/7·49-s − 2.82·50-s − 2.74·53-s + 2.15·55-s + 1.56·59-s + 1.02·61-s + 10.1·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.935568413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.935568413\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 104 T^{3} + 334 T^{4} - 104 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^3$ | \( 1 - 526 T^{4} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 200 T^{3} + 1246 T^{4} - 200 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 2282 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6411 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 44 T^{3} - 2462 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} + 156 T^{3} - 2473 T^{4} + 156 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 174 T^{2} + 11939 T^{4} - 174 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1580 T^{3} + 11806 T^{4} + 1580 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 852 T^{3} + 9938 T^{4} - 852 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 18 T + 212 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 10 T + 50 T^{2} + 100 T^{3} - 3521 T^{4} + 100 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 21018 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 29706 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1700 T^{3} + 14434 T^{4} + 1700 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 10 T + 192 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.17375586450283923429726108540, −6.90842230140822120499082726155, −6.84907533361466127455852495639, −6.47169796527118263132842335370, −6.10501552807487518872148893921, −6.10493285974461232821835100528, −5.66868525508449572952508839069, −5.61394476433454150993086069758, −5.47204185015736159317087445330, −5.15920003972097108055202100366, −4.77082982932744787131488253711, −4.60177308389216887421123301944, −3.82597650809635659707638937230, −3.75627943981277175626955534079, −3.65691613669291354768337129283, −3.49232942677597391224088407902, −3.04440372772912046739631515881, −2.68843977059285732254792068327, −2.38431552496353031339739528707, −2.14431871393003410585023588665, −1.67076141726905268868912326216, −1.40409397905757976555982375723, −1.34415129619409828675840338386, −0.73940129900689794477459905047, −0.67840345039972789167744680862,
0.67840345039972789167744680862, 0.73940129900689794477459905047, 1.34415129619409828675840338386, 1.40409397905757976555982375723, 1.67076141726905268868912326216, 2.14431871393003410585023588665, 2.38431552496353031339739528707, 2.68843977059285732254792068327, 3.04440372772912046739631515881, 3.49232942677597391224088407902, 3.65691613669291354768337129283, 3.75627943981277175626955534079, 3.82597650809635659707638937230, 4.60177308389216887421123301944, 4.77082982932744787131488253711, 5.15920003972097108055202100366, 5.47204185015736159317087445330, 5.61394476433454150993086069758, 5.66868525508449572952508839069, 6.10493285974461232821835100528, 6.10501552807487518872148893921, 6.47169796527118263132842335370, 6.84907533361466127455852495639, 6.90842230140822120499082726155, 7.17375586450283923429726108540
