| L(s) = 1 | + 0.656·2-s − 3-s − 1.56·4-s + 3.35·5-s − 0.656·6-s − 2.47·7-s − 2.34·8-s + 9-s + 2.20·10-s − 1.78·11-s + 1.56·12-s + 0.0701·13-s − 1.62·14-s − 3.35·15-s + 1.59·16-s + 6.97·17-s + 0.656·18-s − 5.06·19-s − 5.25·20-s + 2.47·21-s − 1.17·22-s − 23-s + 2.34·24-s + 6.24·25-s + 0.0461·26-s − 27-s + 3.88·28-s + ⋯ |
| L(s) = 1 | + 0.464·2-s − 0.577·3-s − 0.784·4-s + 1.49·5-s − 0.268·6-s − 0.937·7-s − 0.828·8-s + 0.333·9-s + 0.696·10-s − 0.539·11-s + 0.452·12-s + 0.0194·13-s − 0.435·14-s − 0.865·15-s + 0.399·16-s + 1.69·17-s + 0.154·18-s − 1.16·19-s − 1.17·20-s + 0.541·21-s − 0.250·22-s − 0.208·23-s + 0.478·24-s + 1.24·25-s + 0.00904·26-s − 0.192·27-s + 0.735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.656T + 2T^{2} \) |
| 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 + 2.47T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 - 0.0701T + 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 9.00T + 37T^{2} \) |
| 41 | \( 1 - 0.277T + 41T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 - 3.59T + 53T^{2} \) |
| 59 | \( 1 + 4.41T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 + 7.48T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 2.48T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.994681079936511884941044914373, −8.071439115463818694514956247715, −6.84579790702429598468905934958, −6.11441138439401751601246558577, −5.53921453956224303107386719932, −5.02232116906989047198240741004, −3.79021579615955626395637010056, −2.92803469312906382951106176024, −1.60990813641415412058063364387, 0,
1.60990813641415412058063364387, 2.92803469312906382951106176024, 3.79021579615955626395637010056, 5.02232116906989047198240741004, 5.53921453956224303107386719932, 6.11441138439401751601246558577, 6.84579790702429598468905934958, 8.071439115463818694514956247715, 8.994681079936511884941044914373
