| L(s) = 1 | + 5.31·2-s + 20.2·4-s − 7·7-s + 65.0·8-s − 25.5·11-s + 64.1·13-s − 37.1·14-s + 183.·16-s − 27.6·17-s + 0.792·19-s − 135.·22-s + 108.·23-s + 340.·26-s − 141.·28-s + 234.·29-s + 129.·31-s + 455.·32-s − 147.·34-s − 38.3·37-s + 4.21·38-s + 403.·41-s − 172.·43-s − 516.·44-s + 577.·46-s + 206.·47-s + 49·49-s + 1.29e3·52-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.52·4-s − 0.377·7-s + 2.87·8-s − 0.700·11-s + 1.36·13-s − 0.710·14-s + 2.86·16-s − 0.395·17-s + 0.00956·19-s − 1.31·22-s + 0.984·23-s + 2.56·26-s − 0.956·28-s + 1.49·29-s + 0.748·31-s + 2.51·32-s − 0.742·34-s − 0.170·37-s + 0.0179·38-s + 1.53·41-s − 0.613·43-s − 1.77·44-s + 1.84·46-s + 0.642·47-s + 0.142·49-s + 3.45·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.243213006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.243213006\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 5.31T + 8T^{2} \) |
| 11 | \( 1 + 25.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 0.792T + 6.85e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 38.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 403.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 172.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 679.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 574.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 515.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 556.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 173.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 79.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 652.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 515.T + 9.12e5T^{2} \) |
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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
−8.963803578973258039897546490844, −8.026104097150566752279428359654, −7.03788255939069495801326352452, −6.36540944683004091643706923464, −5.72067491471512878030882738415, −4.83195022623660735580372114330, −4.08569014619025360307097717146, −3.15244472071290912085924819197, −2.48978318033295068395850156511, −1.11157058906844825224662316095,
1.11157058906844825224662316095, 2.48978318033295068395850156511, 3.15244472071290912085924819197, 4.08569014619025360307097717146, 4.83195022623660735580372114330, 5.72067491471512878030882738415, 6.36540944683004091643706923464, 7.03788255939069495801326352452, 8.026104097150566752279428359654, 8.963803578973258039897546490844
