| L(s) = 1 | − 2-s + 2.80·3-s + 4-s − 3.41·5-s − 2.80·6-s + 7-s − 8-s + 4.86·9-s + 3.41·10-s + 2.08·11-s + 2.80·12-s − 5.92·13-s − 14-s − 9.57·15-s + 16-s − 4.64·17-s − 4.86·18-s − 7.52·19-s − 3.41·20-s + 2.80·21-s − 2.08·22-s + 1.02·23-s − 2.80·24-s + 6.64·25-s + 5.92·26-s + 5.23·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.61·3-s + 0.5·4-s − 1.52·5-s − 1.14·6-s + 0.377·7-s − 0.353·8-s + 1.62·9-s + 1.07·10-s + 0.629·11-s + 0.809·12-s − 1.64·13-s − 0.267·14-s − 2.47·15-s + 0.250·16-s − 1.12·17-s − 1.14·18-s − 1.72·19-s − 0.763·20-s + 0.612·21-s − 0.444·22-s + 0.213·23-s − 0.572·24-s + 1.32·25-s + 1.16·26-s + 1.00·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.622357654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.622357654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 + 7.52T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 - 7.40T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 - 4.93T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 + 1.15T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 - 5.22T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 9.99T + 97T^{2} \) |
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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
−8.181384463297239089310834290012, −7.67404915718891324380216373892, −6.99780141371834119917330744348, −6.45077164037989125609259890759, −4.66682208302025308852157157571, −4.36843646302091698533053918191, −3.55263012162705950095273679937, −2.54491970063576186579648420128, −2.16862690247986794798848065734, −0.65981930849464893179481694394,
0.65981930849464893179481694394, 2.16862690247986794798848065734, 2.54491970063576186579648420128, 3.55263012162705950095273679937, 4.36843646302091698533053918191, 4.66682208302025308852157157571, 6.45077164037989125609259890759, 6.99780141371834119917330744348, 7.67404915718891324380216373892, 8.181384463297239089310834290012
