| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 2·5-s + 4·6-s + 3·9-s − 4·10-s − 4·11-s − 2·12-s + 6·13-s − 4·15-s + 16-s + 8·17-s − 6·18-s + 2·20-s + 8·22-s + 2·23-s − 7·25-s − 12·26-s − 4·27-s − 18·29-s + 8·30-s − 4·31-s + 2·32-s + 8·33-s − 16·34-s + 3·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s + 9-s − 1.26·10-s − 1.20·11-s − 0.577·12-s + 1.66·13-s − 1.03·15-s + 1/4·16-s + 1.94·17-s − 1.41·18-s + 0.447·20-s + 1.70·22-s + 0.417·23-s − 7/5·25-s − 2.35·26-s − 0.769·27-s − 3.34·29-s + 1.46·30-s − 0.718·31-s + 0.353·32-s + 1.39·33-s − 2.74·34-s + 1/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 18 T + 137 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 105 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 10 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 135 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 135 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 170 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 145 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76996086458666879592511795441, −7.75275145995101495903977584035, −7.29804932027584102211462132163, −6.95006355049645310986961856266, −6.47993232990247850815679307052, −5.90760455578709863554633863045, −5.74650159126184745197119887211, −5.60157204610417436071006045398, −5.20005055125239507742300537575, −5.07429678409579512922236193469, −3.91553066474489064364651345319, −3.78629826959178660090902051366, −3.74116516308320432692022280776, −2.87427296301054287901307470610, −2.27163411229101606362769056832, −1.86889777355681597143620743459, −1.19221521510694718531444833709, −1.12543954085578698689542637075, 0, 0,
1.12543954085578698689542637075, 1.19221521510694718531444833709, 1.86889777355681597143620743459, 2.27163411229101606362769056832, 2.87427296301054287901307470610, 3.74116516308320432692022280776, 3.78629826959178660090902051366, 3.91553066474489064364651345319, 5.07429678409579512922236193469, 5.20005055125239507742300537575, 5.60157204610417436071006045398, 5.74650159126184745197119887211, 5.90760455578709863554633863045, 6.47993232990247850815679307052, 6.95006355049645310986961856266, 7.29804932027584102211462132163, 7.75275145995101495903977584035, 7.76996086458666879592511795441
