| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 2·5-s − 4·6-s + 4·7-s + 2·8-s + 9-s + 4·10-s − 12·11-s + 2·12-s − 8·14-s − 4·15-s − 4·16-s + 8·17-s − 2·18-s − 2·19-s − 2·20-s + 8·21-s + 24·22-s + 2·23-s + 4·24-s + 25-s − 2·27-s + 4·28-s − 4·29-s + 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s + 1.51·7-s + 0.707·8-s + 1/3·9-s + 1.26·10-s − 3.61·11-s + 0.577·12-s − 2.13·14-s − 1.03·15-s − 16-s + 1.94·17-s − 0.471·18-s − 0.458·19-s − 0.447·20-s + 1.74·21-s + 5.11·22-s + 0.417·23-s + 0.816·24-s + 1/5·25-s − 0.384·27-s + 0.755·28-s − 0.742·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3022377962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3022377962\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 20 T^{2} - 80 T^{3} + 559 T^{4} - 80 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 19 T^{2} + 2 p T^{3} - 4 p T^{4} + 2 p^{2} T^{5} - 19 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} - 200 T^{3} - 1241 T^{4} - 200 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - T^{2} + 270 T^{3} - 1788 T^{4} + 270 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 368 T^{3} - 2501 T^{4} + 368 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} - 144 T^{3} + 2259 T^{4} - 144 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10 T - 25 T^{2} - 190 T^{3} + 7516 T^{4} - 190 p T^{5} - 25 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 184 T^{2} + 1880 T^{3} + 18535 T^{4} + 1880 p T^{5} + 184 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 80 T^{2} + 1911 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 8 T - 40 T^{2} - 304 T^{3} + 1231 T^{4} - 304 p T^{5} - 40 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 52 T^{2} + 928 T^{3} + 17599 T^{4} + 928 p T^{5} + 52 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T - 122 T^{2} - 80 T^{3} + 11539 T^{4} - 80 p T^{5} - 122 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 145 T^{2} - 18 T^{3} + 21636 T^{4} - 18 p T^{5} - 145 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 32 T + 580 T^{2} - 8000 T^{3} + 88399 T^{4} - 8000 p T^{5} + 580 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.75446987128051780598462143209, −7.69473949047785222056410605547, −7.62440058200120831370906751720, −7.47026537422130300335110121314, −7.10580364137254428817306725166, −6.67663650933266709147450144050, −6.47721109688871474794801523110, −5.97681808692294597622231134292, −5.75584529183925547781741132149, −5.35255967979779837355835584778, −5.19595966259254896273159400494, −5.10916755571389116293082246517, −4.92197487259708444174753577423, −4.37973104396228211221740234972, −4.30379635883708533493032749221, −3.88434408569891315821077984489, −3.50442520325816704486954840895, −3.22182525093087833679656898344, −2.75565949865289139121808060381, −2.71529160883312091550446797708, −2.36383698899111400902639438441, −2.00783485135530071835925684591, −1.29917329398976417970806965830, −1.19695066608824315690095506392, −0.21731006135595366352541085317,
0.21731006135595366352541085317, 1.19695066608824315690095506392, 1.29917329398976417970806965830, 2.00783485135530071835925684591, 2.36383698899111400902639438441, 2.71529160883312091550446797708, 2.75565949865289139121808060381, 3.22182525093087833679656898344, 3.50442520325816704486954840895, 3.88434408569891315821077984489, 4.30379635883708533493032749221, 4.37973104396228211221740234972, 4.92197487259708444174753577423, 5.10916755571389116293082246517, 5.19595966259254896273159400494, 5.35255967979779837355835584778, 5.75584529183925547781741132149, 5.97681808692294597622231134292, 6.47721109688871474794801523110, 6.67663650933266709147450144050, 7.10580364137254428817306725166, 7.47026537422130300335110121314, 7.62440058200120831370906751720, 7.69473949047785222056410605547, 7.75446987128051780598462143209
