L(s) = 1 | − 5.58·2-s − 3·3-s + 23.1·4-s + 16.7·6-s + 34.4·7-s − 84.4·8-s + 9·9-s + 11·11-s − 69.4·12-s − 71.2·13-s − 192.·14-s + 286.·16-s − 22.3·17-s − 50.2·18-s + 88.1·19-s − 103.·21-s − 61.3·22-s + 21.5·23-s + 253.·24-s + 397.·26-s − 27·27-s + 796.·28-s + 118.·29-s − 33.5·31-s − 922.·32-s − 33·33-s + 124.·34-s + ⋯ |
L(s) = 1 | − 1.97·2-s − 0.577·3-s + 2.89·4-s + 1.13·6-s + 1.85·7-s − 3.73·8-s + 0.333·9-s + 0.301·11-s − 1.67·12-s − 1.52·13-s − 3.66·14-s + 4.47·16-s − 0.318·17-s − 0.657·18-s + 1.06·19-s − 1.07·21-s − 0.594·22-s + 0.195·23-s + 2.15·24-s + 3.00·26-s − 0.192·27-s + 5.37·28-s + 0.757·29-s − 0.194·31-s − 5.09·32-s − 0.174·33-s + 0.629·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8159450383\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8159450383\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + 5.58T + 8T^{2} \) |
| 7 | \( 1 - 34.4T + 343T^{2} \) |
| 13 | \( 1 + 71.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 21.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 118.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 33.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 364.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 48.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 95.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 132.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 654.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 772.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 112.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 548.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 559.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 48.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 447.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 552.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 413.T + 9.12e5T^{2} \) |
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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
−9.724089459185828051383257295463, −9.099768175153757729023806226210, −7.998539673549190821846537814376, −7.63391554062660546129665909537, −6.81518898857025075379888858869, −5.65213583134590282714099030161, −4.66269092249398591508081657611, −2.65187938368645091686635942805, −1.64896379206934432668667643199, −0.72722087383990481031178273660,
0.72722087383990481031178273660, 1.64896379206934432668667643199, 2.65187938368645091686635942805, 4.66269092249398591508081657611, 5.65213583134590282714099030161, 6.81518898857025075379888858869, 7.63391554062660546129665909537, 7.998539673549190821846537814376, 9.099768175153757729023806226210, 9.724089459185828051383257295463