| L(s) = 1 | + 2-s − 2·3-s + 2·4-s + 5-s − 2·6-s − 4·7-s + 5·8-s − 3·9-s + 10-s − 3·11-s − 4·12-s − 5·13-s − 4·14-s − 2·15-s + 5·16-s − 6·17-s − 3·18-s + 7·19-s + 2·20-s + 8·21-s − 3·22-s − 14·23-s − 10·24-s − 5·26-s + 14·27-s − 8·28-s − 7·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 4-s + 0.447·5-s − 0.816·6-s − 1.51·7-s + 1.76·8-s − 9-s + 0.316·10-s − 0.904·11-s − 1.15·12-s − 1.38·13-s − 1.06·14-s − 0.516·15-s + 5/4·16-s − 1.45·17-s − 0.707·18-s + 1.60·19-s + 0.447·20-s + 1.74·21-s − 0.639·22-s − 2.91·23-s − 2.04·24-s − 0.980·26-s + 2.69·27-s − 1.51·28-s − 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8227246014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8227246014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.04810506722079392610604168601, −10.28319571736789455632240517658, −9.872036998041582077472852435303, −9.839595035518194324831741362933, −9.273232199948768071947077071139, −8.268828255590503060168776330654, −8.145648580987681945901113129016, −7.44571934435379874629893617307, −7.07312929641254898814967058733, −6.50941044628014587976451666950, −6.23406030329819193729561259224, −5.79013511825916754156805582084, −5.26597203143933976397524961780, −5.11333476582095494855558285947, −4.29756212507897062564521025234, −3.77977546925852383595195678130, −2.87991705554582719859113337328, −2.51173710233023751336438617104, −2.03822747743073103949035430428, −0.42499045854351675994417798783,
0.42499045854351675994417798783, 2.03822747743073103949035430428, 2.51173710233023751336438617104, 2.87991705554582719859113337328, 3.77977546925852383595195678130, 4.29756212507897062564521025234, 5.11333476582095494855558285947, 5.26597203143933976397524961780, 5.79013511825916754156805582084, 6.23406030329819193729561259224, 6.50941044628014587976451666950, 7.07312929641254898814967058733, 7.44571934435379874629893617307, 8.145648580987681945901113129016, 8.268828255590503060168776330654, 9.273232199948768071947077071139, 9.839595035518194324831741362933, 9.872036998041582077472852435303, 10.28319571736789455632240517658, 11.04810506722079392610604168601
