| L(s) = 1 | + 3-s − 5-s + 5·7-s − 11-s − 15-s + 8·17-s + 4·19-s + 5·21-s − 4·23-s + 5·25-s − 27-s − 10·29-s − 7·31-s − 33-s − 5·35-s − 8·37-s + 8·41-s + 20·43-s − 6·47-s + 18·49-s + 8·51-s + 53-s + 55-s + 4·57-s − 9·59-s + 2·61-s − 2·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.88·7-s − 0.301·11-s − 0.258·15-s + 1.94·17-s + 0.917·19-s + 1.09·21-s − 0.834·23-s + 25-s − 0.192·27-s − 1.85·29-s − 1.25·31-s − 0.174·33-s − 0.845·35-s − 1.31·37-s + 1.24·41-s + 3.04·43-s − 0.875·47-s + 18/7·49-s + 1.12·51-s + 0.137·53-s + 0.134·55-s + 0.529·57-s − 1.17·59-s + 0.256·61-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.878969584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.878969584\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−10.90552615910957382383820093259, −10.35235380394846928489151366286, −9.785016100286678386750167419489, −9.345334870789155658096372631790, −8.981685868759529814087585806359, −8.492767421029756162863363909018, −7.999251690257549049178788099190, −7.60650591330513169942825525092, −7.50119166433043228449097027836, −7.16758654060059061429937255597, −6.01703606236360138508945314822, −5.68446161154154989611828484702, −5.24967951568593484517241445908, −4.86726850298778436145200389917, −4.08151647151838229731299447282, −3.72124833491832631026741659429, −3.14432682860494934749926411712, −2.33324262068158988677351796615, −1.71915971563612065907288644781, −0.964645434429502727272145962107,
0.964645434429502727272145962107, 1.71915971563612065907288644781, 2.33324262068158988677351796615, 3.14432682860494934749926411712, 3.72124833491832631026741659429, 4.08151647151838229731299447282, 4.86726850298778436145200389917, 5.24967951568593484517241445908, 5.68446161154154989611828484702, 6.01703606236360138508945314822, 7.16758654060059061429937255597, 7.50119166433043228449097027836, 7.60650591330513169942825525092, 7.999251690257549049178788099190, 8.492767421029756162863363909018, 8.981685868759529814087585806359, 9.345334870789155658096372631790, 9.785016100286678386750167419489, 10.35235380394846928489151366286, 10.90552615910957382383820093259
