| L(s) = 1 | − 4·5-s + 6·7-s + 4·11-s + 7·13-s + 8·19-s + 4·23-s + 5·25-s + 4·29-s − 2·31-s − 24·35-s + 14·37-s + 4·41-s + 7·43-s − 2·47-s + 13·49-s − 4·53-s − 16·55-s + 6·59-s + 61-s − 28·65-s + 3·67-s − 2·71-s + 3·73-s + 24·77-s + 5·79-s − 24·83-s + 18·89-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2.26·7-s + 1.20·11-s + 1.94·13-s + 1.83·19-s + 0.834·23-s + 25-s + 0.742·29-s − 0.359·31-s − 4.05·35-s + 2.30·37-s + 0.624·41-s + 1.06·43-s − 0.291·47-s + 13/7·49-s − 0.549·53-s − 2.15·55-s + 0.781·59-s + 0.128·61-s − 3.47·65-s + 0.366·67-s − 0.237·71-s + 0.351·73-s + 2.73·77-s + 0.562·79-s − 2.63·83-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.998519116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.998519116\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.964352994183173528559712572281, −8.425846009219178909549921596084, −8.173944345849538357623051596160, −8.018101391684494754588556326015, −7.52995208028942778429812674112, −7.36535517180210490083423337798, −6.90768362418068808129872231207, −6.19503066982407263835650117920, −6.09581380973770532665644574178, −5.40237620230378471237568583622, −5.03398695111569728729471972275, −4.63727952274961801821900700077, −4.05957846440704100496657405348, −4.04678939340508269137047507859, −3.57165129799234370174013546857, −3.00675521656240391288960224568, −2.42247961769237520801921498526, −1.33301844048870895814082791917, −1.32468323427010767689153182458, −0.814222771209684650277867962788,
0.814222771209684650277867962788, 1.32468323427010767689153182458, 1.33301844048870895814082791917, 2.42247961769237520801921498526, 3.00675521656240391288960224568, 3.57165129799234370174013546857, 4.04678939340508269137047507859, 4.05957846440704100496657405348, 4.63727952274961801821900700077, 5.03398695111569728729471972275, 5.40237620230378471237568583622, 6.09581380973770532665644574178, 6.19503066982407263835650117920, 6.90768362418068808129872231207, 7.36535517180210490083423337798, 7.52995208028942778429812674112, 8.018101391684494754588556326015, 8.173944345849538357623051596160, 8.425846009219178909549921596084, 8.964352994183173528559712572281
