| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s + 6·13-s − 2·15-s − 2·17-s − 4·19-s − 21-s − 25-s − 27-s − 2·29-s − 8·31-s + 2·35-s − 6·37-s − 6·39-s − 10·41-s + 4·43-s + 2·45-s + 8·47-s + 49-s + 2·51-s − 6·53-s + 4·57-s + 4·59-s − 10·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.986·37-s − 0.960·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s − 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.986310439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986310439\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.24403732512986, −12.90899016512909, −12.31458186840173, −11.81250222228826, −11.24428401400436, −10.86340124358859, −10.48034471610787, −10.15497325928369, −9.275391664489580, −8.966276560021113, −8.653980934121771, −7.906041158592371, −7.418670523154315, −6.754951814969638, −6.272289206817513, −5.944589351816773, −5.467673421671065, −4.901789074592131, −4.305610889539971, −3.694689532440865, −3.265213336943514, −2.186574080676432, −1.839037275558300, −1.342767602566739, −0.4187115699163210,
0.4187115699163210, 1.342767602566739, 1.839037275558300, 2.186574080676432, 3.265213336943514, 3.694689532440865, 4.305610889539971, 4.901789074592131, 5.467673421671065, 5.944589351816773, 6.272289206817513, 6.754951814969638, 7.418670523154315, 7.906041158592371, 8.653980934121771, 8.966276560021113, 9.275391664489580, 10.15497325928369, 10.48034471610787, 10.86340124358859, 11.24428401400436, 11.81250222228826, 12.31458186840173, 12.90899016512909, 13.24403732512986
