| L(s) = 1 | + 2-s + 2.18·3-s + 4-s − 1.87·5-s + 2.18·6-s − 4.09·7-s + 8-s + 1.75·9-s − 1.87·10-s + 3.34·11-s + 2.18·12-s + 1.93·13-s − 4.09·14-s − 4.09·15-s + 16-s + 4.44·17-s + 1.75·18-s − 0.0992·19-s − 1.87·20-s − 8.94·21-s + 3.34·22-s + 9.52·23-s + 2.18·24-s − 1.46·25-s + 1.93·26-s − 2.71·27-s − 4.09·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.25·3-s + 0.5·4-s − 0.840·5-s + 0.890·6-s − 1.54·7-s + 0.353·8-s + 0.585·9-s − 0.594·10-s + 1.00·11-s + 0.629·12-s + 0.537·13-s − 1.09·14-s − 1.05·15-s + 0.250·16-s + 1.07·17-s + 0.414·18-s − 0.0227·19-s − 0.420·20-s − 1.95·21-s + 0.712·22-s + 1.98·23-s + 0.445·24-s − 0.293·25-s + 0.380·26-s − 0.521·27-s − 0.774·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.670745691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.670745691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 + 1.87T + 5T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 1.93T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 0.0992T + 19T^{2} \) |
| 23 | \( 1 - 9.52T + 23T^{2} \) |
| 29 | \( 1 - 0.774T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 + 7.11T + 47T^{2} \) |
| 53 | \( 1 - 0.818T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 - 0.658T + 83T^{2} \) |
| 89 | \( 1 - 3.52T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−8.778798329037478682147162013860, −8.081533476460503885943155389472, −7.25268705321476696691460020360, −6.60501949005461800778816629143, −5.87355435992735240763755695126, −4.60272744519567110015619708250, −3.65910843082896368476659406599, −3.33947242051495399610783675521, −2.65875190408075277290824652207, −1.05888475461042723287318390004,
1.05888475461042723287318390004, 2.65875190408075277290824652207, 3.33947242051495399610783675521, 3.65910843082896368476659406599, 4.60272744519567110015619708250, 5.87355435992735240763755695126, 6.60501949005461800778816629143, 7.25268705321476696691460020360, 8.081533476460503885943155389472, 8.778798329037478682147162013860
