| L(s) = 1 | + 2·5-s − 4·7-s − 8·11-s − 4·13-s − 12·19-s − 2·23-s + 4·25-s − 2·29-s + 12·31-s − 8·35-s + 12·37-s + 10·41-s + 12·43-s − 14·47-s − 4·49-s − 4·53-s − 16·55-s + 14·61-s − 8·65-s − 4·67-s + 16·71-s − 14·73-s + 32·77-s − 8·79-s + 16·91-s − 24·95-s + 18·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 2.41·11-s − 1.10·13-s − 2.75·19-s − 0.417·23-s + 4/5·25-s − 0.371·29-s + 2.15·31-s − 1.35·35-s + 1.97·37-s + 1.56·41-s + 1.82·43-s − 2.04·47-s − 4/7·49-s − 0.549·53-s − 2.15·55-s + 1.79·61-s − 0.992·65-s − 0.488·67-s + 1.89·71-s − 1.63·73-s + 3.64·77-s − 0.900·79-s + 1.67·91-s − 2.46·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.975151149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975151149\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T + 12 T^{3} - 29 T^{4} + 12 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 12 p T^{5} - 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 48 T^{2} - 12 T^{3} + 1747 T^{4} - 12 p T^{5} - 48 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 10 T + 21 T^{2} + 30 T^{3} + 460 T^{4} + 30 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 12 T + 50 T^{2} - 96 T^{3} + 795 T^{4} - 96 p T^{5} + 50 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 14 T + 60 T^{2} + 588 T^{3} + 7075 T^{4} + 588 p T^{5} + 60 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 18 T^{2} - 432 T^{3} - 3653 T^{4} - 432 p T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 55 T^{2} - 456 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14 T + 88 T^{2} + 196 T^{3} - 4013 T^{4} + 196 p T^{5} + 88 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T - 115 T^{2} - 12 T^{3} + 11600 T^{4} - 12 p T^{5} - 115 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 16 T + 78 T^{2} - 576 T^{3} + 8467 T^{4} - 576 p T^{5} + 78 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 14 T + 29 T^{2} + 294 T^{3} + 8252 T^{4} + 294 p T^{5} + 29 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 18 T + 77 T^{2} - 954 T^{3} + 20172 T^{4} - 954 p T^{5} + 77 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−6.27625408463510717474122726260, −5.99644108475227599309039960351, −5.94978859697278165471046718646, −5.71807633932060588040453969209, −5.54221969841503472916907689276, −5.26256474916239944741246221604, −4.92137631006050160954480695273, −4.81840677818229608780173884392, −4.62001640863046373575699539564, −4.35755023439305227804017246180, −4.31840373019551003757016778795, −4.12103743839861962788306122632, −3.62724726668245021473998402965, −3.30579099035312938934230556953, −3.14192159089391962099632388997, −3.04483043459968440243080596193, −2.52759750500203615321565477079, −2.50785538549592776473395199656, −2.37493411796785675003607466866, −2.24676868777830501788122684679, −1.71474862467648282091139088676, −1.61448955139252002146015092117, −0.72826437299051796752591450648, −0.59453285689327340530651658940, −0.34794401117931944115246479489,
0.34794401117931944115246479489, 0.59453285689327340530651658940, 0.72826437299051796752591450648, 1.61448955139252002146015092117, 1.71474862467648282091139088676, 2.24676868777830501788122684679, 2.37493411796785675003607466866, 2.50785538549592776473395199656, 2.52759750500203615321565477079, 3.04483043459968440243080596193, 3.14192159089391962099632388997, 3.30579099035312938934230556953, 3.62724726668245021473998402965, 4.12103743839861962788306122632, 4.31840373019551003757016778795, 4.35755023439305227804017246180, 4.62001640863046373575699539564, 4.81840677818229608780173884392, 4.92137631006050160954480695273, 5.26256474916239944741246221604, 5.54221969841503472916907689276, 5.71807633932060588040453969209, 5.94978859697278165471046718646, 5.99644108475227599309039960351, 6.27625408463510717474122726260
