| L(s) = 1 | − 0.177·2-s + 8.83·3-s − 7.96·4-s + 5·5-s − 1.57·6-s + 5.98·7-s + 2.83·8-s + 51.0·9-s − 0.888·10-s − 11·11-s − 70.3·12-s + 61.0·13-s − 1.06·14-s + 44.1·15-s + 63.2·16-s + 49.5·17-s − 9.07·18-s + 19·19-s − 39.8·20-s + 52.8·21-s + 1.95·22-s − 45.5·23-s + 25.0·24-s + 25·25-s − 10.8·26-s + 212.·27-s − 47.6·28-s + ⋯ |
| L(s) = 1 | − 0.0628·2-s + 1.69·3-s − 0.996·4-s + 0.447·5-s − 0.106·6-s + 0.323·7-s + 0.125·8-s + 1.88·9-s − 0.0281·10-s − 0.301·11-s − 1.69·12-s + 1.30·13-s − 0.0203·14-s + 0.760·15-s + 0.988·16-s + 0.707·17-s − 0.118·18-s + 0.229·19-s − 0.445·20-s + 0.549·21-s + 0.0189·22-s − 0.413·23-s + 0.213·24-s + 0.200·25-s − 0.0818·26-s + 1.51·27-s − 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.884797861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.884797861\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 0.177T + 8T^{2} \) |
| 3 | \( 1 - 8.83T + 27T^{2} \) |
| 7 | \( 1 - 5.98T + 343T^{2} \) |
| 13 | \( 1 - 61.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 49.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 45.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 63.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 85.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 315.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 482.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 678.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 790.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 487.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 365.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 962.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−9.332637716387971911866124292183, −8.756507857626885992601563365792, −8.070751301859878459914505516939, −7.50580687313102600237685675708, −6.07634681793762340818134392986, −5.06496149442549518670835299393, −3.91690945927753434484731744927, −3.37938741949025413391340908440, −2.11693133552598101023793787667, −1.05800755165426815350070766597,
1.05800755165426815350070766597, 2.11693133552598101023793787667, 3.37938741949025413391340908440, 3.91690945927753434484731744927, 5.06496149442549518670835299393, 6.07634681793762340818134392986, 7.50580687313102600237685675708, 8.070751301859878459914505516939, 8.756507857626885992601563365792, 9.332637716387971911866124292183
