| L(s) = 1 | + 2-s + 2.66·3-s + 4-s + 3.10·5-s + 2.66·6-s − 1.64·7-s + 8-s + 4.11·9-s + 3.10·10-s + 3.30·11-s + 2.66·12-s + 2.59·13-s − 1.64·14-s + 8.27·15-s + 16-s + 1.50·17-s + 4.11·18-s − 2.71·19-s + 3.10·20-s − 4.38·21-s + 3.30·22-s + 6.22·23-s + 2.66·24-s + 4.63·25-s + 2.59·26-s + 2.97·27-s − 1.64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 0.5·4-s + 1.38·5-s + 1.08·6-s − 0.621·7-s + 0.353·8-s + 1.37·9-s + 0.981·10-s + 0.996·11-s + 0.769·12-s + 0.719·13-s − 0.439·14-s + 2.13·15-s + 0.250·16-s + 0.364·17-s + 0.969·18-s − 0.623·19-s + 0.694·20-s − 0.956·21-s + 0.704·22-s + 1.29·23-s + 0.544·24-s + 0.926·25-s + 0.508·26-s + 0.571·27-s − 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.660037143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.660037143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 - 1.50T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 + 4.40T + 29T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 + 2.25T + 47T^{2} \) |
| 53 | \( 1 + 7.37T + 53T^{2} \) |
| 59 | \( 1 - 1.40T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 3.82T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
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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
−8.148995713388129162113466556335, −7.28889471030687307416465178616, −6.43186858411161541204128631924, −6.19369424353599382824243566935, −5.15052695835972340664855666472, −4.28074488336870228738393934841, −3.26751463433542357461477121649, −3.11600661815690912731280333736, −1.93114278137243635611198063363, −1.47728751917006691519651450817,
1.47728751917006691519651450817, 1.93114278137243635611198063363, 3.11600661815690912731280333736, 3.26751463433542357461477121649, 4.28074488336870228738393934841, 5.15052695835972340664855666472, 6.19369424353599382824243566935, 6.43186858411161541204128631924, 7.28889471030687307416465178616, 8.148995713388129162113466556335
