L(s) = 1 | − 2.69·2-s + 3.34·3-s + 5.27·4-s + 2.36·5-s − 9.02·6-s − 0.269·7-s − 8.83·8-s + 8.19·9-s − 6.39·10-s − 3.44·11-s + 17.6·12-s + 0.0670·13-s + 0.728·14-s + 7.92·15-s + 13.2·16-s + 0.574·17-s − 22.0·18-s + 5.60·19-s + 12.5·20-s − 0.902·21-s + 9.28·22-s + 0.0744·23-s − 29.5·24-s + 0.613·25-s − 0.180·26-s + 17.3·27-s − 1.42·28-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 1.93·3-s + 2.63·4-s + 1.05·5-s − 3.68·6-s − 0.102·7-s − 3.12·8-s + 2.73·9-s − 2.02·10-s − 1.03·11-s + 5.09·12-s + 0.0186·13-s + 0.194·14-s + 2.04·15-s + 3.32·16-s + 0.139·17-s − 5.20·18-s + 1.28·19-s + 2.79·20-s − 0.197·21-s + 1.97·22-s + 0.0155·23-s − 6.03·24-s + 0.122·25-s − 0.0354·26-s + 3.34·27-s − 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.390808492\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.390808492\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 503 | \( 1 - T \) |
good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 - 3.34T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 + 0.269T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 - 0.0670T + 13T^{2} \) |
| 17 | \( 1 - 0.574T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 0.0744T + 23T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 + 0.941T + 41T^{2} \) |
| 43 | \( 1 - 0.827T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 0.0750T + 59T^{2} \) |
| 61 | \( 1 - 8.44T + 61T^{2} \) |
| 67 | \( 1 - 5.59T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 3.81T + 79T^{2} \) |
| 83 | \( 1 + 4.05T + 83T^{2} \) |
| 89 | \( 1 + 6.39T + 89T^{2} \) |
| 97 | \( 1 - 7.52T + 97T^{2} \) |
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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
−10.25482693063714125452999654906, −9.633737746368473803327534039628, −9.322434175947800378304213060499, −8.380409887635310411633915473124, −7.64248307622211245160004138247, −7.08919504960939889406737650465, −5.63932161938421165563344518939, −3.34617194568261289342658080920, −2.40165868216749761707793980879, −1.60271385325966885598636133335,
1.60271385325966885598636133335, 2.40165868216749761707793980879, 3.34617194568261289342658080920, 5.63932161938421165563344518939, 7.08919504960939889406737650465, 7.64248307622211245160004138247, 8.380409887635310411633915473124, 9.322434175947800378304213060499, 9.633737746368473803327534039628, 10.25482693063714125452999654906