| L(s) = 1 | − 2·3-s + 3·9-s − 8·19-s − 6·25-s − 4·27-s + 12·29-s + 16·31-s + 12·37-s − 7·49-s − 4·53-s + 16·57-s + 8·59-s + 12·75-s + 5·81-s − 8·83-s − 24·87-s − 32·93-s + 32·103-s − 4·109-s − 24·111-s + 36·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 14·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.83·19-s − 6/5·25-s − 0.769·27-s + 2.22·29-s + 2.87·31-s + 1.97·37-s − 49-s − 0.549·53-s + 2.11·57-s + 1.04·59-s + 1.38·75-s + 5/9·81-s − 0.878·83-s − 2.57·87-s − 3.31·93-s + 3.15·103-s − 0.383·109-s − 2.27·111-s + 3.38·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.15·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8788731802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8788731802\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.964671915727173343349228861429, −9.935819229981957589664665865525, −8.959311313448748386566246803110, −8.304042274266532161103040174092, −8.098990694093691505068092710868, −7.35977874808944660694472335106, −6.42897107072744719896577758454, −6.41480629060511724242965007307, −5.98894980010644060477023411018, −5.02364207902997377499660058417, −4.44992200701182918033448908949, −4.25303028692796488061253338187, −3.00889317555517420664706578512, −2.17837496002353596178446785744, −0.852162344962028721020593862686,
0.852162344962028721020593862686, 2.17837496002353596178446785744, 3.00889317555517420664706578512, 4.25303028692796488061253338187, 4.44992200701182918033448908949, 5.02364207902997377499660058417, 5.98894980010644060477023411018, 6.41480629060511724242965007307, 6.42897107072744719896577758454, 7.35977874808944660694472335106, 8.098990694093691505068092710868, 8.304042274266532161103040174092, 8.959311313448748386566246803110, 9.935819229981957589664665865525, 9.964671915727173343349228861429
