| L(s) = 1 | − 2·3-s + 4-s + 3·7-s + 9-s − 2·12-s + 13-s + 16-s + 13·19-s − 6·21-s + 8·25-s + 4·27-s + 3·28-s − 5·31-s + 36-s + 4·37-s − 2·39-s − 8·43-s − 2·48-s + 52-s − 26·57-s + 7·61-s + 3·63-s + 64-s − 5·67-s − 26·73-s − 16·75-s + 13·76-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1.13·7-s + 1/3·9-s − 0.577·12-s + 0.277·13-s + 1/4·16-s + 2.98·19-s − 1.30·21-s + 8/5·25-s + 0.769·27-s + 0.566·28-s − 0.898·31-s + 1/6·36-s + 0.657·37-s − 0.320·39-s − 1.21·43-s − 0.288·48-s + 0.138·52-s − 3.44·57-s + 0.896·61-s + 0.377·63-s + 1/8·64-s − 0.610·67-s − 3.04·73-s − 1.84·75-s + 1.49·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143892 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143892 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.545895059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545895059\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 571 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.272762618559898070625948853561, −8.866027480477555413181549144883, −8.275403852094367136220801912757, −7.76904822222023173185877262189, −7.16979594458889325760104610039, −7.03622876715536341177055025138, −6.24861667502802432205911086548, −5.63957052298413472319460572104, −5.36011133515852555771491600985, −4.89412023042970398531111158902, −4.33759753882143354737063600703, −3.25209923447453440263568928674, −2.92289552295651634986330875961, −1.62486327942013293986723731830, −1.02026844426092223925184591038,
1.02026844426092223925184591038, 1.62486327942013293986723731830, 2.92289552295651634986330875961, 3.25209923447453440263568928674, 4.33759753882143354737063600703, 4.89412023042970398531111158902, 5.36011133515852555771491600985, 5.63957052298413472319460572104, 6.24861667502802432205911086548, 7.03622876715536341177055025138, 7.16979594458889325760104610039, 7.76904822222023173185877262189, 8.275403852094367136220801912757, 8.866027480477555413181549144883, 9.272762618559898070625948853561
