| L(s) = 1 | + 2-s − 2.56·3-s + 4-s + 5-s − 2.56·6-s + 3.71·7-s + 8-s + 3.56·9-s + 10-s − 11-s − 2.56·12-s + 5.57·13-s + 3.71·14-s − 2.56·15-s + 16-s + 3.56·17-s + 3.56·18-s + 7.50·19-s + 20-s − 9.51·21-s − 22-s − 3.57·23-s − 2.56·24-s + 25-s + 5.57·26-s − 1.45·27-s + 3.71·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.47·3-s + 0.5·4-s + 0.447·5-s − 1.04·6-s + 1.40·7-s + 0.353·8-s + 1.18·9-s + 0.316·10-s − 0.301·11-s − 0.739·12-s + 1.54·13-s + 0.992·14-s − 0.661·15-s + 0.250·16-s + 0.865·17-s + 0.840·18-s + 1.72·19-s + 0.223·20-s − 2.07·21-s − 0.213·22-s − 0.746·23-s − 0.523·24-s + 0.200·25-s + 1.09·26-s − 0.279·27-s + 0.701·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.917064083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.917064083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 - 7.50T + 19T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 + 1.18T + 37T^{2} \) |
| 41 | \( 1 - 9.18T + 41T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 9.37T + 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 6.61T + 67T^{2} \) |
| 71 | \( 1 + 0.469T + 71T^{2} \) |
| 73 | \( 1 + 1.64T + 73T^{2} \) |
| 79 | \( 1 + 4.43T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 0.511T + 97T^{2} \) |
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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
−7.956025303016115457145403627699, −7.57156544957932290468107472732, −6.39907136659760764864025381384, −6.03140685239265504667261248223, −5.18790578242403276907541975149, −5.02822315820220277643281862145, −4.00486338420961002866546391450, −3.02585719999520149622429646144, −1.59248687664043637863577571798, −1.05869620729122783900593829843,
1.05869620729122783900593829843, 1.59248687664043637863577571798, 3.02585719999520149622429646144, 4.00486338420961002866546391450, 5.02822315820220277643281862145, 5.18790578242403276907541975149, 6.03140685239265504667261248223, 6.39907136659760764864025381384, 7.57156544957932290468107472732, 7.956025303016115457145403627699
