| L(s) = 1 | + 2-s + 0.279·3-s + 4-s + 1.09·5-s + 0.279·6-s + 1.36·7-s + 8-s − 2.92·9-s + 1.09·10-s + 11-s + 0.279·12-s + 4.66·13-s + 1.36·14-s + 0.305·15-s + 16-s − 3.42·17-s − 2.92·18-s + 3.93·19-s + 1.09·20-s + 0.380·21-s + 22-s − 0.253·23-s + 0.279·24-s − 3.80·25-s + 4.66·26-s − 1.65·27-s + 1.36·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.161·3-s + 0.5·4-s + 0.488·5-s + 0.114·6-s + 0.514·7-s + 0.353·8-s − 0.973·9-s + 0.345·10-s + 0.301·11-s + 0.0807·12-s + 1.29·13-s + 0.363·14-s + 0.0788·15-s + 0.250·16-s − 0.831·17-s − 0.688·18-s + 0.902·19-s + 0.244·20-s + 0.0829·21-s + 0.213·22-s − 0.0528·23-s + 0.0570·24-s − 0.761·25-s + 0.915·26-s − 0.318·27-s + 0.257·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.958862555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.958862555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.279T + 3T^{2} \) |
| 5 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 + 3.42T + 17T^{2} \) |
| 19 | \( 1 - 3.93T + 19T^{2} \) |
| 23 | \( 1 + 0.253T + 23T^{2} \) |
| 29 | \( 1 - 8.46T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 0.397T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 - 1.93T + 53T^{2} \) |
| 59 | \( 1 + 4.37T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 4.27T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.313339480814077330305270769860, −7.71713769095708284770418352113, −6.64908968099120540833594498684, −6.06379657206656406381338413062, −5.51606788639338812424277709423, −4.61990870997091609607332250095, −3.83036279687592340105559784224, −2.95599664987858975795018879618, −2.13612290380942726162017884474, −1.05855350439227160326883303966,
1.05855350439227160326883303966, 2.13612290380942726162017884474, 2.95599664987858975795018879618, 3.83036279687592340105559784224, 4.61990870997091609607332250095, 5.51606788639338812424277709423, 6.06379657206656406381338413062, 6.64908968099120540833594498684, 7.71713769095708284770418352113, 8.313339480814077330305270769860
