| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s − 2·9-s + 2·11-s − 2·12-s + 8·13-s − 4·16-s + 4·18-s − 4·22-s − 2·23-s − 9·25-s − 16·26-s + 5·27-s + 8·32-s − 2·33-s − 4·36-s + 6·37-s − 8·39-s + 4·44-s + 4·46-s + 16·47-s + 4·48-s − 10·49-s + 18·50-s + 16·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 2/3·9-s + 0.603·11-s − 0.577·12-s + 2.21·13-s − 16-s + 0.942·18-s − 0.852·22-s − 0.417·23-s − 9/5·25-s − 3.13·26-s + 0.962·27-s + 1.41·32-s − 0.348·33-s − 2/3·36-s + 0.986·37-s − 1.28·39-s + 0.603·44-s + 0.589·46-s + 2.33·47-s + 0.577·48-s − 1.42·49-s + 2.54·50-s + 2.21·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4650245394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4650245394\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−11.21532888591095488055645771466, −10.24006992909568321138419784955, −10.03550909718107888433464868208, −9.314280432061735820103645414759, −8.603539619290756001226038948684, −8.557305878080159620602459979711, −7.85816077117362959855929652794, −7.17975828892938553190702219115, −6.36261389471308870138602900888, −6.05614842101165534491399200306, −5.39918512850282004974717248783, −4.19169950645670655455560288935, −3.65854567503247488502961160345, −2.23405840566606208508614377911, −1.00859600183935355798315238789,
1.00859600183935355798315238789, 2.23405840566606208508614377911, 3.65854567503247488502961160345, 4.19169950645670655455560288935, 5.39918512850282004974717248783, 6.05614842101165534491399200306, 6.36261389471308870138602900888, 7.17975828892938553190702219115, 7.85816077117362959855929652794, 8.557305878080159620602459979711, 8.603539619290756001226038948684, 9.314280432061735820103645414759, 10.03550909718107888433464868208, 10.24006992909568321138419784955, 11.21532888591095488055645771466
