| L(s) = 1 | − 2.53·2-s + 1.42·3-s + 4.42·4-s − 0.780·5-s − 3.60·6-s + 3.78·7-s − 6.15·8-s − 0.977·9-s + 1.97·10-s + 2.96·11-s + 6.29·12-s + 1.57·13-s − 9.60·14-s − 1.11·15-s + 6.74·16-s + 2.53·17-s + 2.47·18-s − 7.71·19-s − 3.45·20-s + 5.38·21-s − 7.50·22-s − 4.09·23-s − 8.75·24-s − 4.39·25-s − 4.00·26-s − 5.65·27-s + 16.7·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.821·3-s + 2.21·4-s − 0.349·5-s − 1.47·6-s + 1.43·7-s − 2.17·8-s − 0.325·9-s + 0.625·10-s + 0.893·11-s + 1.81·12-s + 0.437·13-s − 2.56·14-s − 0.286·15-s + 1.68·16-s + 0.615·17-s + 0.583·18-s − 1.76·19-s − 0.772·20-s + 1.17·21-s − 1.60·22-s − 0.853·23-s − 1.78·24-s − 0.878·25-s − 0.784·26-s − 1.08·27-s + 3.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 - 1.42T + 3T^{2} \) |
| 5 | \( 1 + 0.780T + 5T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 + 7.71T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 0.435T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 + 0.989T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 0.490T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 + 9.88T + 61T^{2} \) |
| 67 | \( 1 - 3.59T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 + 2.41T + 89T^{2} \) |
| 97 | \( 1 - 4.04T + 97T^{2} \) |
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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
−7.75989275071493958542513782207, −7.35344774762177866397001669681, −6.28711148892275155403395761760, −5.78415159180250182513810527164, −4.40750957205759901754257991175, −3.80170992955907415869293224782, −2.65321165809617041176894356772, −1.90562781814065499421040995152, −1.34846678997573612465444664642, 0,
1.34846678997573612465444664642, 1.90562781814065499421040995152, 2.65321165809617041176894356772, 3.80170992955907415869293224782, 4.40750957205759901754257991175, 5.78415159180250182513810527164, 6.28711148892275155403395761760, 7.35344774762177866397001669681, 7.75989275071493958542513782207
