| L(s) = 1 | − 0.238·2-s + 3-s − 1.94·4-s + 2.19·5-s − 0.238·6-s + 5.28·7-s + 0.939·8-s + 9-s − 0.523·10-s + 4.75·11-s − 1.94·12-s − 13-s − 1.25·14-s + 2.19·15-s + 3.66·16-s + 2.71·17-s − 0.238·18-s − 6.38·19-s − 4.26·20-s + 5.28·21-s − 1.13·22-s + 8.68·23-s + 0.939·24-s − 0.178·25-s + 0.238·26-s + 27-s − 10.2·28-s + ⋯ |
| L(s) = 1 | − 0.168·2-s + 0.577·3-s − 0.971·4-s + 0.981·5-s − 0.0972·6-s + 1.99·7-s + 0.332·8-s + 0.333·9-s − 0.165·10-s + 1.43·11-s − 0.560·12-s − 0.277·13-s − 0.336·14-s + 0.566·15-s + 0.915·16-s + 0.657·17-s − 0.0561·18-s − 1.46·19-s − 0.954·20-s + 1.15·21-s − 0.241·22-s + 1.81·23-s + 0.191·24-s − 0.0357·25-s + 0.0467·26-s + 0.192·27-s − 1.93·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.154786980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.154786980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 0.238T + 2T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 7 | \( 1 - 5.28T + 7T^{2} \) |
| 11 | \( 1 - 4.75T + 11T^{2} \) |
| 17 | \( 1 - 2.71T + 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 - 4.85T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 1.83T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 0.824T + 67T^{2} \) |
| 71 | \( 1 + 5.55T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 3.13T + 79T^{2} \) |
| 83 | \( 1 + 4.17T + 83T^{2} \) |
| 89 | \( 1 - 4.45T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.706274036331645581221222919565, −7.87158723977286531220617554586, −7.21021355187727153503664073942, −6.13033086047341594688970140792, −5.33387930658804775167883703752, −4.52869024959944539542239585106, −4.17760953584962894170151194229, −2.84352635647488375795725828583, −1.64078915529363106346677719344, −1.24841763475556005185825239346,
1.24841763475556005185825239346, 1.64078915529363106346677719344, 2.84352635647488375795725828583, 4.17760953584962894170151194229, 4.52869024959944539542239585106, 5.33387930658804775167883703752, 6.13033086047341594688970140792, 7.21021355187727153503664073942, 7.87158723977286531220617554586, 8.706274036331645581221222919565
