L(s) = 1 | − 2·3-s − 3·4-s − 4·5-s + 3·9-s − 11-s + 6·12-s + 8·15-s + 5·16-s + 12·20-s + 2·25-s − 4·27-s + 16·31-s + 2·33-s − 9·36-s + 12·37-s + 3·44-s − 12·45-s − 16·47-s − 10·48-s + 49-s + 12·53-s + 4·55-s + 8·59-s − 24·60-s − 3·64-s − 24·67-s − 4·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 3/2·4-s − 1.78·5-s + 9-s − 0.301·11-s + 1.73·12-s + 2.06·15-s + 5/4·16-s + 2.68·20-s + 2/5·25-s − 0.769·27-s + 2.87·31-s + 0.348·33-s − 3/2·36-s + 1.97·37-s + 0.452·44-s − 1.78·45-s − 2.33·47-s − 1.44·48-s + 1/7·49-s + 1.64·53-s + 0.539·55-s + 1.04·59-s − 3.09·60-s − 3/8·64-s − 2.93·67-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3470684500\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3470684500\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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
−8.146329212122427589249395398868, −8.141517002480388222220952641794, −7.61386400817600432579143653551, −7.21790373610916871268434804799, −6.49973685708325266933280216312, −6.06507120539099918049903981653, −5.68299786063791157253646201739, −4.78033654342364343647378297963, −4.60121025692501797340970242340, −4.49476807110266902405624429159, −3.69478633591386645327522922963, −3.39460536306959657073087615016, −2.40997607300375870079584083833, −1.02008449562817059954899874886, −0.42828310674610679900828276088,
0.42828310674610679900828276088, 1.02008449562817059954899874886, 2.40997607300375870079584083833, 3.39460536306959657073087615016, 3.69478633591386645327522922963, 4.49476807110266902405624429159, 4.60121025692501797340970242340, 4.78033654342364343647378297963, 5.68299786063791157253646201739, 6.06507120539099918049903981653, 6.49973685708325266933280216312, 7.21790373610916871268434804799, 7.61386400817600432579143653551, 8.141517002480388222220952641794, 8.146329212122427589249395398868