| L(s) = 1 | + 3-s + 4-s − 2·7-s + 9-s + 12-s − 4·13-s + 16-s + 8·19-s − 2·21-s + 25-s + 27-s − 2·28-s + 36-s + 12·37-s − 4·39-s − 8·43-s + 48-s + 3·49-s − 4·52-s + 8·57-s + 28·61-s − 2·63-s + 64-s − 24·67-s + 20·73-s + 75-s + 8·76-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 1.83·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 1/6·36-s + 1.97·37-s − 0.640·39-s − 1.21·43-s + 0.144·48-s + 3/7·49-s − 0.554·52-s + 1.05·57-s + 3.58·61-s − 0.251·63-s + 1/8·64-s − 2.93·67-s + 2.34·73-s + 0.115·75-s + 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.041868430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041868430\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.437870818355534879168031071986, −9.118954839776025628038155273468, −8.185634457079877915713969006222, −7.983671143656830403181428322838, −7.31970008287706573052341981271, −7.07941601012087388140383379015, −6.44305888127139518988332786048, −5.97975998469911468351052676550, −5.09599254318882530046900067399, −4.95725936124254114017607791343, −3.85422726721423423764474467794, −3.44893990823261081236809071688, −2.69678884580615531725956839791, −2.27051973415844153304852765225, −0.996088059894723745963619169753,
0.996088059894723745963619169753, 2.27051973415844153304852765225, 2.69678884580615531725956839791, 3.44893990823261081236809071688, 3.85422726721423423764474467794, 4.95725936124254114017607791343, 5.09599254318882530046900067399, 5.97975998469911468351052676550, 6.44305888127139518988332786048, 7.07941601012087388140383379015, 7.31970008287706573052341981271, 7.983671143656830403181428322838, 8.185634457079877915713969006222, 9.118954839776025628038155273468, 9.437870818355534879168031071986
