| L(s) = 1 | − 3-s + 3·7-s + 9-s − 11-s − 13-s + 3·17-s + 2·19-s − 3·21-s + 5·23-s − 27-s − 6·29-s − 10·31-s + 33-s − 5·37-s + 39-s + 3·41-s + 4·43-s + 6·47-s + 2·49-s − 3·51-s − 5·53-s − 2·57-s + 8·59-s + 61-s + 3·63-s + 12·67-s − 5·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s − 0.654·21-s + 1.04·23-s − 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.174·33-s − 0.821·37-s + 0.160·39-s + 0.468·41-s + 0.609·43-s + 0.875·47-s + 2/7·49-s − 0.420·51-s − 0.686·53-s − 0.264·57-s + 1.04·59-s + 0.128·61-s + 0.377·63-s + 1.46·67-s − 0.601·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.963105381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963105381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−16.17584545719260, −15.32036034284402, −14.96708651082813, −14.29355498303952, −13.97826433311375, −13.01051515942928, −12.68176708970636, −12.05983747307476, −11.34648955157331, −11.03296185050650, −10.56795704661969, −9.708318391869949, −9.240228048047062, −8.503614198214375, −7.768570703463606, −7.359329244895325, −6.799971233144237, −5.706008568586367, −5.377767878388566, −4.895856588658951, −4.010735522347763, −3.343063461043710, −2.272633253062288, −1.558370744918726, −0.6516608020654029,
0.6516608020654029, 1.558370744918726, 2.272633253062288, 3.343063461043710, 4.010735522347763, 4.895856588658951, 5.377767878388566, 5.706008568586367, 6.799971233144237, 7.359329244895325, 7.768570703463606, 8.503614198214375, 9.240228048047062, 9.708318391869949, 10.56795704661969, 11.03296185050650, 11.34648955157331, 12.05983747307476, 12.68176708970636, 13.01051515942928, 13.97826433311375, 14.29355498303952, 14.96708651082813, 15.32036034284402, 16.17584545719260
