L(s) = 1 | − 2-s + 1.04·3-s + 4-s − 0.929·5-s − 1.04·6-s + 2.89·7-s − 8-s − 1.90·9-s + 0.929·10-s − 3.97·11-s + 1.04·12-s − 2.89·14-s − 0.972·15-s + 16-s − 4.54·17-s + 1.90·18-s + 19-s − 0.929·20-s + 3.02·21-s + 3.97·22-s − 4.09·23-s − 1.04·24-s − 4.13·25-s − 5.13·27-s + 2.89·28-s + 1.64·29-s + 0.972·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.604·3-s + 0.5·4-s − 0.415·5-s − 0.427·6-s + 1.09·7-s − 0.353·8-s − 0.635·9-s + 0.293·10-s − 1.19·11-s + 0.302·12-s − 0.773·14-s − 0.251·15-s + 0.250·16-s − 1.10·17-s + 0.449·18-s + 0.229·19-s − 0.207·20-s + 0.660·21-s + 0.847·22-s − 0.854·23-s − 0.213·24-s − 0.827·25-s − 0.987·27-s + 0.546·28-s + 0.306·29-s + 0.177·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.223531968\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.223531968\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 + 0.929T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 + 3.97T + 11T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 37 | \( 1 + 2.85T + 37T^{2} \) |
| 41 | \( 1 - 0.453T + 41T^{2} \) |
| 43 | \( 1 - 4.16T + 43T^{2} \) |
| 47 | \( 1 - 9.08T + 47T^{2} \) |
| 53 | \( 1 + 3.63T + 53T^{2} \) |
| 59 | \( 1 + 9.02T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 9.60T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 + 6.23T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−7.975322744788202928093267250682, −7.84459216722634129975253437157, −6.82210697219729678466892902752, −5.94522658801332790074153935866, −5.16568390761214396832679288690, −4.39505499904900281789757874781, −3.46069656164720733674886794342, −2.43076610808118839351044852542, −2.06268637061304900643775013977, −0.58942988568268433531400093651,
0.58942988568268433531400093651, 2.06268637061304900643775013977, 2.43076610808118839351044852542, 3.46069656164720733674886794342, 4.39505499904900281789757874781, 5.16568390761214396832679288690, 5.94522658801332790074153935866, 6.82210697219729678466892902752, 7.84459216722634129975253437157, 7.975322744788202928093267250682