| L(s) = 1 | − 2.64·2-s − 3-s + 4.99·4-s − 3.91·5-s + 2.64·6-s − 7-s − 7.90·8-s + 9-s + 10.3·10-s + 4.79·11-s − 4.99·12-s + 2.79·13-s + 2.64·14-s + 3.91·15-s + 10.9·16-s + 0.382·17-s − 2.64·18-s + 1.51·19-s − 19.5·20-s + 21-s − 12.6·22-s − 3.70·23-s + 7.90·24-s + 10.3·25-s − 7.38·26-s − 27-s − 4.99·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.49·4-s − 1.74·5-s + 1.07·6-s − 0.377·7-s − 2.79·8-s + 0.333·9-s + 3.27·10-s + 1.44·11-s − 1.44·12-s + 0.774·13-s + 0.706·14-s + 1.01·15-s + 2.73·16-s + 0.0927·17-s − 0.623·18-s + 0.346·19-s − 4.36·20-s + 0.218·21-s − 2.70·22-s − 0.772·23-s + 1.61·24-s + 2.06·25-s − 1.44·26-s − 0.192·27-s − 0.943·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4634867950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4634867950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 5 | \( 1 + 3.91T + 5T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 - 0.382T + 17T^{2} \) |
| 19 | \( 1 - 1.51T + 19T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 6.54T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 0.724T + 67T^{2} \) |
| 71 | \( 1 + 0.0715T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 2.10T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 9.45T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.560896619991909606820581218535, −7.72404632588597703030408390408, −7.32825434652431325677755273921, −6.42623168788210423970534684353, −6.10502468684730232370193017391, −4.43823499658076828858423920128, −3.75899178231937771641457155856, −2.78698899404871805391151567582, −1.25336582536099686065500313194, −0.62579451509590989717990820956,
0.62579451509590989717990820956, 1.25336582536099686065500313194, 2.78698899404871805391151567582, 3.75899178231937771641457155856, 4.43823499658076828858423920128, 6.10502468684730232370193017391, 6.42623168788210423970534684353, 7.32825434652431325677755273921, 7.72404632588597703030408390408, 8.560896619991909606820581218535
