| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s + 2·7-s − 4·8-s + 3·9-s + 4·10-s − 6·12-s − 9·13-s − 4·14-s + 4·15-s + 5·16-s + 17-s − 6·18-s + 2·19-s − 6·20-s − 4·21-s + 7·23-s + 8·24-s + 3·25-s + 18·26-s − 4·27-s + 6·28-s + 3·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 1.26·10-s − 1.73·12-s − 2.49·13-s − 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.242·17-s − 1.41·18-s + 0.458·19-s − 1.34·20-s − 0.872·21-s + 1.45·23-s + 1.63·24-s + 3/5·25-s + 3.53·26-s − 0.769·27-s + 1.13·28-s + 0.557·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4860670629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4860670629\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 9 T + 45 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 47 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 59 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 67 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 15 T + 129 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 11 T + 85 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 159 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 129 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T - 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.575877150701891649083416692445, −8.482112172739504241655941987941, −7.74941461330178274104848994311, −7.73510543053223778155682464593, −7.28430555222644324657491798790, −6.98393926749935122470583570288, −6.69318836150357271561055732475, −6.52595582974709996887642585523, −5.51568677556932875540569360388, −5.24183036563599671736682081842, −5.11240328260593920082804794093, −4.88629643928981562046459857205, −3.93728582106357798979979787420, −3.88380330093231725045555668489, −2.85069428754552180810750142096, −2.85038648297873167871488539937, −1.88241483933277554569135058843, −1.69554895581313174170166439735, −0.75051979530605171493580269027, −0.41768616197908533919644723895,
0.41768616197908533919644723895, 0.75051979530605171493580269027, 1.69554895581313174170166439735, 1.88241483933277554569135058843, 2.85038648297873167871488539937, 2.85069428754552180810750142096, 3.88380330093231725045555668489, 3.93728582106357798979979787420, 4.88629643928981562046459857205, 5.11240328260593920082804794093, 5.24183036563599671736682081842, 5.51568677556932875540569360388, 6.52595582974709996887642585523, 6.69318836150357271561055732475, 6.98393926749935122470583570288, 7.28430555222644324657491798790, 7.73510543053223778155682464593, 7.74941461330178274104848994311, 8.482112172739504241655941987941, 8.575877150701891649083416692445
