| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 2·13-s + 15-s + 6·17-s + 2·19-s − 4·21-s − 23-s + 25-s + 27-s + 6·29-s − 4·31-s − 4·35-s + 8·37-s + 2·39-s + 6·41-s + 8·43-s + 45-s + 12·47-s + 9·49-s + 6·51-s − 6·53-s + 2·57-s − 6·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.458·19-s − 0.872·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.676·35-s + 1.31·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 0.264·57-s − 0.781·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.045648760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045648760\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.511758896372669857590428269448, −9.030659518610676735272651166608, −7.921326128817937118162276358597, −7.21054166557932961770913706263, −6.17142169768220094960595771652, −5.70505981480309089659266266120, −4.26729899831963658544162449634, −3.30616925826625651856196563306, −2.64835837043768983129661084204, −1.05969550603591767149922029580,
1.05969550603591767149922029580, 2.64835837043768983129661084204, 3.30616925826625651856196563306, 4.26729899831963658544162449634, 5.70505981480309089659266266120, 6.17142169768220094960595771652, 7.21054166557932961770913706263, 7.921326128817937118162276358597, 9.030659518610676735272651166608, 9.511758896372669857590428269448
