L(s) = 1 | + 1.44·2-s + 3-s + 0.0928·4-s + 2.29·5-s + 1.44·6-s + 7-s − 2.75·8-s + 9-s + 3.32·10-s − 1.55·11-s + 0.0928·12-s + 6.79·13-s + 1.44·14-s + 2.29·15-s − 4.17·16-s + 0.345·17-s + 1.44·18-s − 2.91·19-s + 0.213·20-s + 21-s − 2.24·22-s + 1.63·23-s − 2.75·24-s + 0.276·25-s + 9.82·26-s + 27-s + 0.0928·28-s + ⋯ |
L(s) = 1 | + 1.02·2-s + 0.577·3-s + 0.0464·4-s + 1.02·5-s + 0.590·6-s + 0.377·7-s − 0.975·8-s + 0.333·9-s + 1.05·10-s − 0.467·11-s + 0.0268·12-s + 1.88·13-s + 0.386·14-s + 0.593·15-s − 1.04·16-s + 0.0838·17-s + 0.340·18-s − 0.668·19-s + 0.0476·20-s + 0.218·21-s − 0.478·22-s + 0.340·23-s − 0.563·24-s + 0.0552·25-s + 1.92·26-s + 0.192·27-s + 0.0175·28-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
\(L(1)\) |
\(\approx\) |
\(4.809192933\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.809192933\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.44T + 2T^{2} \) |
| 5 | \( 1 - 2.29T + 5T^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 13 | \( 1 - 6.79T + 13T^{2} \) |
| 17 | \( 1 - 0.345T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 + 0.433T + 29T^{2} \) |
| 31 | \( 1 - 4.80T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 1.20T + 43T^{2} \) |
| 47 | \( 1 + 0.954T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 - 8.57T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 - 1.62T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6.27T + 83T^{2} \) |
| 89 | \( 1 + 8.34T + 89T^{2} \) |
| 97 | \( 1 + 2.81T + 97T^{2} \) |
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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
−8.534439609870125402838309180713, −7.83654032394565910386348653374, −6.64730286641231509181652173859, −5.99578147824496270485118491687, −5.57159535460392000877447580793, −4.53317491252451341412564818334, −3.97408520032292252815270022461, −3.01817068420627175647003112848, −2.27549011241920247807518820975, −1.13729999947655940400882125400,
1.13729999947655940400882125400, 2.27549011241920247807518820975, 3.01817068420627175647003112848, 3.97408520032292252815270022461, 4.53317491252451341412564818334, 5.57159535460392000877447580793, 5.99578147824496270485118491687, 6.64730286641231509181652173859, 7.83654032394565910386348653374, 8.534439609870125402838309180713