L(s) = 1 | − 2.75·2-s + 1.84·3-s + 5.59·4-s − 0.0856·5-s − 5.07·6-s − 0.236·7-s − 9.92·8-s + 0.393·9-s + 0.236·10-s − 2.52·11-s + 10.3·12-s − 0.243·13-s + 0.650·14-s − 0.157·15-s + 16.1·16-s − 17-s − 1.08·18-s + 0.722·19-s − 0.479·20-s − 0.434·21-s + 6.94·22-s − 7.44·23-s − 18.2·24-s − 4.99·25-s + 0.670·26-s − 4.80·27-s − 1.32·28-s + ⋯ |
L(s) = 1 | − 1.94·2-s + 1.06·3-s + 2.79·4-s − 0.0382·5-s − 2.07·6-s − 0.0892·7-s − 3.50·8-s + 0.131·9-s + 0.0746·10-s − 0.759·11-s + 2.97·12-s − 0.0675·13-s + 0.173·14-s − 0.0407·15-s + 4.03·16-s − 0.242·17-s − 0.255·18-s + 0.165·19-s − 0.107·20-s − 0.0949·21-s + 1.48·22-s − 1.55·23-s − 3.73·24-s − 0.998·25-s + 0.131·26-s − 0.923·27-s − 0.249·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 + 0.0856T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 13 | \( 1 + 0.243T + 13T^{2} \) |
| 19 | \( 1 - 0.722T + 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 1.99T + 29T^{2} \) |
| 31 | \( 1 - 0.670T + 31T^{2} \) |
| 37 | \( 1 + 8.56T + 37T^{2} \) |
| 41 | \( 1 + 2.34T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 - 6.85T + 53T^{2} \) |
| 61 | \( 1 - 2.52T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 + 2.04T + 71T^{2} \) |
| 73 | \( 1 - 9.29T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 6.73T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−9.552964156888012895719233647602, −8.602320707273185936323691014442, −8.134149121892202346640068548731, −7.56503432987964762860469900237, −6.59658445954847543756767556780, −5.59498487764225963522418354149, −3.63140894875287382459502496574, −2.59804878129978935780534933981, −1.83843138836059449826730125288, 0,
1.83843138836059449826730125288, 2.59804878129978935780534933981, 3.63140894875287382459502496574, 5.59498487764225963522418354149, 6.59658445954847543756767556780, 7.56503432987964762860469900237, 8.134149121892202346640068548731, 8.602320707273185936323691014442, 9.552964156888012895719233647602