L(s) = 1 | + 4·3-s − 4·7-s + 6·9-s + 4·19-s − 16·21-s − 2·25-s − 4·27-s + 12·29-s − 16·31-s − 4·37-s + 9·49-s + 12·53-s + 16·57-s + 12·59-s − 24·63-s − 8·75-s − 37·81-s − 12·83-s + 48·87-s − 64·93-s + 8·103-s + 28·109-s − 16·111-s − 36·113-s + 10·121-s + 127-s + 131-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 1.51·7-s + 2·9-s + 0.917·19-s − 3.49·21-s − 2/5·25-s − 0.769·27-s + 2.22·29-s − 2.87·31-s − 0.657·37-s + 9/7·49-s + 1.64·53-s + 2.11·57-s + 1.56·59-s − 3.02·63-s − 0.923·75-s − 4.11·81-s − 1.31·83-s + 5.14·87-s − 6.63·93-s + 0.788·103-s + 2.68·109-s − 1.51·111-s − 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.724747304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.724747304\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−13.85527597205343874413781727946, −13.51800823632316294200121117071, −13.08341381244233473952115796668, −12.59967805565372363074940228804, −11.95841262022681953421076004057, −11.34442682559592549094610972239, −10.45584994758087072117626657560, −9.921506810920154637386781506468, −9.507527997314415692233235404029, −9.003599649400614938492986071576, −8.614121972701083607506245823139, −8.128628623235270634301562672712, −7.20708683661451296405070797755, −7.11598076117609151613707720329, −6.00442836075500351053765247795, −5.36827972142785832925545706293, −3.99977327571498627519619112045, −3.45534973712552675432088105877, −2.96568982120453400149957970307, −2.17971886719975111308459041619,
2.17971886719975111308459041619, 2.96568982120453400149957970307, 3.45534973712552675432088105877, 3.99977327571498627519619112045, 5.36827972142785832925545706293, 6.00442836075500351053765247795, 7.11598076117609151613707720329, 7.20708683661451296405070797755, 8.128628623235270634301562672712, 8.614121972701083607506245823139, 9.003599649400614938492986071576, 9.507527997314415692233235404029, 9.921506810920154637386781506468, 10.45584994758087072117626657560, 11.34442682559592549094610972239, 11.95841262022681953421076004057, 12.59967805565372363074940228804, 13.08341381244233473952115796668, 13.51800823632316294200121117071, 13.85527597205343874413781727946