| L(s) = 1 | + 1.28·2-s − 6.34·4-s + 12.4·5-s − 9.00·7-s − 18.4·8-s + 16.0·10-s − 51.0·11-s − 11.5·14-s + 27.0·16-s − 6.86·17-s − 83.1·19-s − 78.9·20-s − 65.7·22-s + 187.·23-s + 29.9·25-s + 57.1·28-s − 223.·29-s + 57.3·31-s + 182.·32-s − 8.82·34-s − 112.·35-s − 156.·37-s − 106.·38-s − 229.·40-s + 222.·41-s + 347.·43-s + 324.·44-s + ⋯ |
| L(s) = 1 | + 0.454·2-s − 0.793·4-s + 1.11·5-s − 0.486·7-s − 0.815·8-s + 0.506·10-s − 1.40·11-s − 0.221·14-s + 0.421·16-s − 0.0978·17-s − 1.00·19-s − 0.882·20-s − 0.636·22-s + 1.70·23-s + 0.239·25-s + 0.385·28-s − 1.43·29-s + 0.332·31-s + 1.00·32-s − 0.0445·34-s − 0.541·35-s − 0.694·37-s − 0.456·38-s − 0.908·40-s + 0.846·41-s + 1.23·43-s + 1.11·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.762804770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.762804770\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.28T + 8T^{2} \) |
| 5 | \( 1 - 12.4T + 125T^{2} \) |
| 7 | \( 1 + 9.00T + 343T^{2} \) |
| 11 | \( 1 + 51.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 6.86T + 4.91e3T^{2} \) |
| 19 | \( 1 + 83.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 187.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 57.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 347.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 45.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 473.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 615.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 195.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 355.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 763.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 207.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 251.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.55e3T + 9.12e5T^{2} \) |
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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
−9.161403392974496886853655043874, −8.506845423764428135140203996221, −7.44819801186163467852480612799, −6.43826753939196332201281253337, −5.57474336201943882984438461587, −5.16544683586659644976798211135, −4.09415811296795647800283719023, −3.00258304103630215619634211508, −2.16307086424769569624798648740, −0.58531447736850017812077306988,
0.58531447736850017812077306988, 2.16307086424769569624798648740, 3.00258304103630215619634211508, 4.09415811296795647800283719023, 5.16544683586659644976798211135, 5.57474336201943882984438461587, 6.43826753939196332201281253337, 7.44819801186163467852480612799, 8.506845423764428135140203996221, 9.161403392974496886853655043874
