| L(s) = 1 | − 2·5-s − 7-s − 5·11-s − 5·13-s + 8·17-s + 4·19-s + 7·23-s + 25-s − 5·29-s − 5·31-s + 2·35-s − 12·37-s − 8·43-s − 5·47-s + 18·53-s + 10·55-s + 26·59-s − 2·61-s + 10·65-s + 14·67-s − 18·71-s + 5·77-s − 6·79-s − 10·83-s − 16·85-s + 6·89-s + 5·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.50·11-s − 1.38·13-s + 1.94·17-s + 0.917·19-s + 1.45·23-s + 1/5·25-s − 0.928·29-s − 0.898·31-s + 0.338·35-s − 1.97·37-s − 1.21·43-s − 0.729·47-s + 2.47·53-s + 1.34·55-s + 3.38·59-s − 0.256·61-s + 1.24·65-s + 1.71·67-s − 2.13·71-s + 0.569·77-s − 0.675·79-s − 1.09·83-s − 1.73·85-s + 0.635·89-s + 0.524·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.793363729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.793363729\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + T + T^{2} - 2 p T^{3} - 8 p T^{4} - 2 p^{2} T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 + 5 T + 7 T^{2} - 40 T^{3} - 170 T^{4} - 40 p T^{5} + 7 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 7 T + 5 T^{2} + 14 T^{3} + 280 T^{4} + 14 p T^{5} + 5 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 5 T - 25 T^{2} - 40 T^{3} + 934 T^{4} - 40 p T^{5} - 25 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 5 T - 29 T^{2} - 40 T^{3} + 1180 T^{4} - 40 p T^{5} - 29 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 25 T^{2} - 1056 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 5 T - 61 T^{2} - 40 T^{3} + 4012 T^{4} - 40 p T^{5} - 61 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 62 T^{2} - 112 T^{3} + 391 T^{4} - 112 p T^{5} - 62 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 70 T^{2} + 112 T^{3} - 1745 T^{4} + 112 p T^{5} + 70 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 74 T^{2} - 288 T^{3} + 3015 T^{4} - 288 p T^{5} - 74 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 10 T - 34 T^{2} - 320 T^{3} + 2767 T^{4} - 320 p T^{5} - 34 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 16 T + 159 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.09925083368134653594092433837, −5.63661864379034776542001094716, −5.44837942339292435963169058332, −5.43590690678611229897868269154, −5.42969412832855644202376512975, −5.28641907115237213210037992604, −5.06014006757891907360464634253, −4.88916031061894270373107541228, −4.31591032022742999178370831275, −4.18108384875981868390701580774, −4.13549531751586077961136025781, −3.93400762030274299023364035412, −3.45758623441384494762995693303, −3.45286782218502572716589536350, −2.97034135266453540496403006162, −2.96663573115802318698406795440, −2.93731378285775187473723937460, −2.63174143404559382185289653467, −2.06123706445758663165016355980, −1.92261902818342044740994526986, −1.79057216632657186770698716915, −1.23001109275533122997346350556, −0.988441127705432430257004561055, −0.39509856336968475628597315738, −0.39213207595486492653136790845,
0.39213207595486492653136790845, 0.39509856336968475628597315738, 0.988441127705432430257004561055, 1.23001109275533122997346350556, 1.79057216632657186770698716915, 1.92261902818342044740994526986, 2.06123706445758663165016355980, 2.63174143404559382185289653467, 2.93731378285775187473723937460, 2.96663573115802318698406795440, 2.97034135266453540496403006162, 3.45286782218502572716589536350, 3.45758623441384494762995693303, 3.93400762030274299023364035412, 4.13549531751586077961136025781, 4.18108384875981868390701580774, 4.31591032022742999178370831275, 4.88916031061894270373107541228, 5.06014006757891907360464634253, 5.28641907115237213210037992604, 5.42969412832855644202376512975, 5.43590690678611229897868269154, 5.44837942339292435963169058332, 5.63661864379034776542001094716, 6.09925083368134653594092433837
