| L(s) = 1 | + 3.43·2-s − 9.61·3-s + 3.79·4-s + 1.55·5-s − 33.0·6-s − 14.4·8-s + 65.4·9-s + 5.33·10-s − 14.6·11-s − 36.4·12-s + 69.9·13-s − 14.9·15-s − 79.9·16-s + 70.7·17-s + 224.·18-s + 87.5·19-s + 5.88·20-s − 50.4·22-s + 75.2·23-s + 139.·24-s − 122.·25-s + 240.·26-s − 370.·27-s − 95.3·29-s − 51.2·30-s + 24.8·31-s − 158.·32-s + ⋯ |
| L(s) = 1 | + 1.21·2-s − 1.85·3-s + 0.473·4-s + 0.138·5-s − 2.24·6-s − 0.638·8-s + 2.42·9-s + 0.168·10-s − 0.402·11-s − 0.876·12-s + 1.49·13-s − 0.257·15-s − 1.24·16-s + 1.00·17-s + 2.94·18-s + 1.05·19-s + 0.0658·20-s − 0.489·22-s + 0.682·23-s + 1.18·24-s − 0.980·25-s + 1.81·26-s − 2.63·27-s − 0.610·29-s − 0.312·30-s + 0.143·31-s − 0.877·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.960866110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.960866110\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 3.43T + 8T^{2} \) |
| 3 | \( 1 + 9.61T + 27T^{2} \) |
| 5 | \( 1 - 1.55T + 125T^{2} \) |
| 11 | \( 1 + 14.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 95.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 24.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 432.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 8.31T + 7.95e4T^{2} \) |
| 47 | \( 1 + 3.55T + 1.03e5T^{2} \) |
| 53 | \( 1 - 82.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 769.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 266.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 10.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 782.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 613.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 427.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 955.T + 9.12e5T^{2} \) |
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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
−8.583437648305234503985739033505, −7.42349296529881004266721167584, −6.66335286760313889864341516262, −5.87982375983828295367353287589, −5.49986377890693460698211925691, −4.94685541863717981772146025670, −3.94025487031852925993573836064, −3.28786424787090970289792197824, −1.60057156960235149929605816225, −0.59612718976148673767904199762,
0.59612718976148673767904199762, 1.60057156960235149929605816225, 3.28786424787090970289792197824, 3.94025487031852925993573836064, 4.94685541863717981772146025670, 5.49986377890693460698211925691, 5.87982375983828295367353287589, 6.66335286760313889864341516262, 7.42349296529881004266721167584, 8.583437648305234503985739033505
