| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 2·11-s − 3·13-s + 15-s + 17-s − 2·19-s − 21-s + 3·23-s + 25-s − 27-s − 4·29-s + 5·31-s + 2·33-s − 35-s + 37-s + 3·39-s + 11·41-s + 4·43-s − 45-s + 9·47-s + 49-s − 51-s + 2·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.832·13-s + 0.258·15-s + 0.242·17-s − 0.458·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 0.898·31-s + 0.348·33-s − 0.169·35-s + 0.164·37-s + 0.480·39-s + 1.71·41-s + 0.609·43-s − 0.149·45-s + 1.31·47-s + 1/7·49-s − 0.140·51-s + 0.269·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343717932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343717932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.69817048409241, −12.82513322877940, −12.73260348952368, −12.12337483924051, −11.74891010300284, −11.02297085673368, −10.89315252132489, −10.30913314966844, −9.768226303818282, −9.187507024661447, −8.747805008299409, −7.996840426160851, −7.576943174649977, −7.307119178799569, −6.603360406172408, −5.969778755321974, −5.515604611884502, −4.974655308374570, −4.330600632260190, −4.132325372025140, −3.085676301062236, −2.646463144603246, −1.947713285983290, −1.077017046312325, −0.4178150229750929,
0.4178150229750929, 1.077017046312325, 1.947713285983290, 2.646463144603246, 3.085676301062236, 4.132325372025140, 4.330600632260190, 4.974655308374570, 5.515604611884502, 5.969778755321974, 6.603360406172408, 7.307119178799569, 7.576943174649977, 7.996840426160851, 8.747805008299409, 9.187507024661447, 9.768226303818282, 10.30913314966844, 10.89315252132489, 11.02297085673368, 11.74891010300284, 12.12337483924051, 12.73260348952368, 12.82513322877940, 13.69817048409241
