| L(s) = 1 | + 2.00·2-s + 3-s + 2.01·4-s + 3.94·5-s + 2.00·6-s − 2.43·7-s + 0.0387·8-s + 9-s + 7.90·10-s + 5.06·11-s + 2.01·12-s + 0.849·13-s − 4.88·14-s + 3.94·15-s − 3.96·16-s − 3.54·17-s + 2.00·18-s + 2.59·19-s + 7.96·20-s − 2.43·21-s + 10.1·22-s − 23-s + 0.0387·24-s + 10.5·25-s + 1.70·26-s + 27-s − 4.92·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1.00·4-s + 1.76·5-s + 0.818·6-s − 0.921·7-s + 0.0136·8-s + 0.333·9-s + 2.50·10-s + 1.52·11-s + 0.582·12-s + 0.235·13-s − 1.30·14-s + 1.01·15-s − 0.990·16-s − 0.859·17-s + 0.472·18-s + 0.594·19-s + 1.78·20-s − 0.531·21-s + 2.16·22-s − 0.208·23-s + 0.00790·24-s + 2.11·25-s + 0.334·26-s + 0.192·27-s − 0.929·28-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
| \(L(1)\) |
\(\approx\) |
\(5.816591829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.816591829\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.00T + 2T^{2} \) |
| 5 | \( 1 - 3.94T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 0.849T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 - 2.59T + 19T^{2} \) |
| 31 | \( 1 + 0.555T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 - 2.45T + 41T^{2} \) |
| 43 | \( 1 + 7.48T + 43T^{2} \) |
| 47 | \( 1 - 4.02T + 47T^{2} \) |
| 53 | \( 1 - 3.73T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 8.57T + 79T^{2} \) |
| 83 | \( 1 - 8.09T + 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 - 0.381T + 97T^{2} \) |
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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
−9.238434775917371504219782710134, −8.739658130026834311163202078120, −7.05651298443967467177624308253, −6.47352447353763762531434444408, −6.01928178361829618734022284308, −5.16412590367761421140204980921, −4.15789956970075750758210612324, −3.36207127112987279444442944543, −2.51692747968992455828458942190, −1.56344667397900133937342656020,
1.56344667397900133937342656020, 2.51692747968992455828458942190, 3.36207127112987279444442944543, 4.15789956970075750758210612324, 5.16412590367761421140204980921, 6.01928178361829618734022284308, 6.47352447353763762531434444408, 7.05651298443967467177624308253, 8.739658130026834311163202078120, 9.238434775917371504219782710134
