| L(s) = 1 | + 3·4-s − 2·11-s + 5·16-s + 2·19-s + 12·29-s + 8·31-s + 10·41-s − 6·44-s + 5·49-s + 22·59-s + 28·61-s + 3·64-s + 10·71-s + 6·76-s − 10·79-s − 20·89-s + 22·101-s + 24·109-s + 36·116-s + 3·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.603·11-s + 5/4·16-s + 0.458·19-s + 2.22·29-s + 1.43·31-s + 1.56·41-s − 0.904·44-s + 5/7·49-s + 2.86·59-s + 3.58·61-s + 3/8·64-s + 1.18·71-s + 0.688·76-s − 1.12·79-s − 2.11·89-s + 2.18·101-s + 2.29·109-s + 3.34·116-s + 3/11·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.852694804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.852694804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−9.017041722458801716994923028904, −8.568891059817690710786278515823, −8.220659555474796358667056744293, −8.148868382659839341113182116194, −7.34048424876465677186142570469, −7.28023622010360050458234235152, −6.79806317192970297973470887521, −6.55760243283327021991045613731, −6.02171445368533873754317562258, −5.77506754366716819444239097594, −5.20833064497245747081884276596, −4.89858419998148600373127449043, −4.29102506892960570404081863746, −3.84998293964607259201682175877, −3.28092545924390481765675384592, −2.73066940417275293644769737710, −2.34941860132668614952611307400, −2.24582127773753597897398860312, −1.04255413523686930824297306454, −0.893115655868234040583118614749,
0.893115655868234040583118614749, 1.04255413523686930824297306454, 2.24582127773753597897398860312, 2.34941860132668614952611307400, 2.73066940417275293644769737710, 3.28092545924390481765675384592, 3.84998293964607259201682175877, 4.29102506892960570404081863746, 4.89858419998148600373127449043, 5.20833064497245747081884276596, 5.77506754366716819444239097594, 6.02171445368533873754317562258, 6.55760243283327021991045613731, 6.79806317192970297973470887521, 7.28023622010360050458234235152, 7.34048424876465677186142570469, 8.148868382659839341113182116194, 8.220659555474796358667056744293, 8.568891059817690710786278515823, 9.017041722458801716994923028904
