| L(s) = 1 | + 2-s + 2.23·3-s + 4-s − 0.404·5-s + 2.23·6-s + 3.10·7-s + 8-s + 2.00·9-s − 0.404·10-s + 0.636·11-s + 2.23·12-s + 6.59·13-s + 3.10·14-s − 0.905·15-s + 16-s − 2.47·17-s + 2.00·18-s + 4.07·19-s − 0.404·20-s + 6.94·21-s + 0.636·22-s + 2.49·23-s + 2.23·24-s − 4.83·25-s + 6.59·26-s − 2.23·27-s + 3.10·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.181·5-s + 0.913·6-s + 1.17·7-s + 0.353·8-s + 0.667·9-s − 0.128·10-s + 0.191·11-s + 0.645·12-s + 1.82·13-s + 0.829·14-s − 0.233·15-s + 0.250·16-s − 0.599·17-s + 0.471·18-s + 0.934·19-s − 0.0905·20-s + 1.51·21-s + 0.135·22-s + 0.520·23-s + 0.456·24-s − 0.967·25-s + 1.29·26-s − 0.429·27-s + 0.586·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.807560407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.807560407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 0.404T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 - 0.636T + 11T^{2} \) |
| 13 | \( 1 - 6.59T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 4.07T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 + 6.03T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 - 7.69T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 3.38T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 - 2.32T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 6.39T + 67T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 5.18T + 83T^{2} \) |
| 89 | \( 1 + 4.79T + 89T^{2} \) |
| 97 | \( 1 + 2.79T + 97T^{2} \) |
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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
−8.352892780560448973670321572114, −7.75020101938150034032817964804, −7.23471505343340111471452402911, −6.05399078414768522219520700726, −5.48039759842417475502557314264, −4.37920242523263522789180650486, −3.80230479650367589165741372660, −3.15043012177826351255505219730, −2.06013837950243326986167299089, −1.37708488658876272250760487378,
1.37708488658876272250760487378, 2.06013837950243326986167299089, 3.15043012177826351255505219730, 3.80230479650367589165741372660, 4.37920242523263522789180650486, 5.48039759842417475502557314264, 6.05399078414768522219520700726, 7.23471505343340111471452402911, 7.75020101938150034032817964804, 8.352892780560448973670321572114
