| L(s) = 1 | + 0.468·2-s + 0.438·3-s − 1.78·4-s + 1.40·5-s + 0.205·6-s + 0.0838·7-s − 1.77·8-s − 2.80·9-s + 0.657·10-s + 3.19·11-s − 0.780·12-s + 0.0393·14-s + 0.614·15-s + 2.72·16-s + 2.18·17-s − 1.31·18-s − 5.42·19-s − 2.49·20-s + 0.0367·21-s + 1.49·22-s − 6.49·23-s − 0.777·24-s − 3.03·25-s − 2.54·27-s − 0.149·28-s + 3.84·29-s + 0.288·30-s + ⋯ |
| L(s) = 1 | + 0.331·2-s + 0.253·3-s − 0.890·4-s + 0.626·5-s + 0.0839·6-s + 0.0316·7-s − 0.626·8-s − 0.935·9-s + 0.207·10-s + 0.962·11-s − 0.225·12-s + 0.0105·14-s + 0.158·15-s + 0.682·16-s + 0.529·17-s − 0.310·18-s − 1.24·19-s − 0.557·20-s + 0.00802·21-s + 0.319·22-s − 1.35·23-s − 0.158·24-s − 0.607·25-s − 0.490·27-s − 0.0282·28-s + 0.714·29-s + 0.0526·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.924739496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924739496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.468T + 2T^{2} \) |
| 3 | \( 1 - 0.438T + 3T^{2} \) |
| 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 - 0.0838T + 7T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 17 | \( 1 - 2.18T + 17T^{2} \) |
| 19 | \( 1 + 5.42T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 3.84T + 29T^{2} \) |
| 37 | \( 1 - 8.45T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 2.31T + 53T^{2} \) |
| 59 | \( 1 - 0.297T + 59T^{2} \) |
| 61 | \( 1 - 7.42T + 61T^{2} \) |
| 67 | \( 1 - 5.01T + 67T^{2} \) |
| 71 | \( 1 + 9.37T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 7.04T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.289643808515959386115807882261, −7.72599334709966564875118370903, −6.36649072790843368277568926099, −6.05348466804887452200997388180, −5.39653304657944806536469226737, −4.30701050579312168289332976730, −3.93983337902287593057801241467, −2.87706938486762944670155774918, −2.03202601754011184862332579233, −0.70257956221964881009974581627,
0.70257956221964881009974581627, 2.03202601754011184862332579233, 2.87706938486762944670155774918, 3.93983337902287593057801241467, 4.30701050579312168289332976730, 5.39653304657944806536469226737, 6.05348466804887452200997388180, 6.36649072790843368277568926099, 7.72599334709966564875118370903, 8.289643808515959386115807882261
