L(s) = 1 | − 9.80·3-s − 18.3·5-s − 24.4·7-s + 69.1·9-s + 2.36·11-s + 17.2·13-s + 179.·15-s − 17·17-s − 136.·19-s + 239.·21-s + 2.04·23-s + 210.·25-s − 413.·27-s + 170.·29-s + 188.·31-s − 23.2·33-s + 447.·35-s − 137.·37-s − 169.·39-s − 260.·41-s − 502.·43-s − 1.26e3·45-s + 199.·47-s + 253.·49-s + 166.·51-s + 128.·53-s − 43.3·55-s + ⋯ |
L(s) = 1 | − 1.88·3-s − 1.63·5-s − 1.31·7-s + 2.56·9-s + 0.0649·11-s + 0.367·13-s + 3.09·15-s − 0.242·17-s − 1.65·19-s + 2.48·21-s + 0.0185·23-s + 1.68·25-s − 2.94·27-s + 1.09·29-s + 1.09·31-s − 0.122·33-s + 2.15·35-s − 0.612·37-s − 0.694·39-s − 0.992·41-s − 1.78·43-s − 4.19·45-s + 0.619·47-s + 0.737·49-s + 0.457·51-s + 0.333·53-s − 0.106·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2725396161\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2725396161\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
good | 3 | \( 1 + 9.80T + 27T^{2} \) |
| 5 | \( 1 + 18.3T + 125T^{2} \) |
| 7 | \( 1 + 24.4T + 343T^{2} \) |
| 11 | \( 1 - 2.36T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.2T + 2.19e3T^{2} \) |
| 19 | \( 1 + 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 2.04T + 1.21e4T^{2} \) |
| 29 | \( 1 - 170.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 260.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 128.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 303.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 351.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 405.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 965.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 110.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 168.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 56.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 574.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 966.T + 9.12e5T^{2} \) |
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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
−12.36666714462234528763508930089, −11.83290619741898809457073359483, −10.86574481796279466049250595659, −10.09561587741969939131074060747, −8.386991273105921086201568295595, −6.85440708211076355822277827073, −6.39200659422848598377453344074, −4.78263713765196885361548173208, −3.74353362366995200048396450763, −0.45793017319916872744750227499,
0.45793017319916872744750227499, 3.74353362366995200048396450763, 4.78263713765196885361548173208, 6.39200659422848598377453344074, 6.85440708211076355822277827073, 8.386991273105921086201568295595, 10.09561587741969939131074060747, 10.86574481796279466049250595659, 11.83290619741898809457073359483, 12.36666714462234528763508930089