| L(s) = 1 | − 1.57·2-s − 5.53·4-s − 6.97·5-s + 21.2·8-s + 10.9·10-s + 32.5·11-s + 19.3·13-s + 10.8·16-s − 22.2·17-s + 155.·19-s + 38.5·20-s − 51.1·22-s + 76.9·23-s − 76.3·25-s − 30.3·26-s − 122.·29-s − 164.·31-s − 187.·32-s + 34.9·34-s + 66.2·37-s − 243.·38-s − 148.·40-s + 231.·41-s − 210.·43-s − 180.·44-s − 120.·46-s − 206.·47-s + ⋯ |
| L(s) = 1 | − 0.555·2-s − 0.691·4-s − 0.623·5-s + 0.939·8-s + 0.346·10-s + 0.892·11-s + 0.412·13-s + 0.169·16-s − 0.317·17-s + 1.87·19-s + 0.431·20-s − 0.495·22-s + 0.697·23-s − 0.611·25-s − 0.229·26-s − 0.781·29-s − 0.954·31-s − 1.03·32-s + 0.176·34-s + 0.294·37-s − 1.04·38-s − 0.585·40-s + 0.883·41-s − 0.746·43-s − 0.617·44-s − 0.387·46-s − 0.640·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.116922426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.116922426\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.57T + 8T^{2} \) |
| 5 | \( 1 + 6.97T + 125T^{2} \) |
| 11 | \( 1 - 32.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 76.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 122.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 66.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 210.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 419.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 300.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 24.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 274.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 336.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 693.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 584.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−9.278152196637518418542515771274, −8.577508963313668464615334444565, −7.65166643134531769653954109203, −7.16587114314631111473304542418, −5.89587606754336161988992472053, −4.97558156978367370865904329592, −4.00826063272024448316888000141, −3.33271733968894316057613519684, −1.59237771399940184716505464579, −0.62227424419970931131937199954,
0.62227424419970931131937199954, 1.59237771399940184716505464579, 3.33271733968894316057613519684, 4.00826063272024448316888000141, 4.97558156978367370865904329592, 5.89587606754336161988992472053, 7.16587114314631111473304542418, 7.65166643134531769653954109203, 8.577508963313668464615334444565, 9.278152196637518418542515771274
