| L(s) = 1 | − 4.90·2-s + 16.0·4-s + 5.61·5-s − 3.21·7-s − 39.4·8-s − 27.5·10-s + 43.6·11-s + 15.7·14-s + 65.0·16-s − 65.8·17-s − 3.06·19-s + 90.0·20-s − 214.·22-s + 163.·23-s − 93.4·25-s − 51.5·28-s − 155.·29-s + 240.·31-s − 3.58·32-s + 323.·34-s − 18.0·35-s − 317.·37-s + 15.0·38-s − 221.·40-s + 127.·41-s − 379.·43-s + 700.·44-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.00·4-s + 0.502·5-s − 0.173·7-s − 1.74·8-s − 0.870·10-s + 1.19·11-s + 0.300·14-s + 1.01·16-s − 0.939·17-s − 0.0369·19-s + 1.00·20-s − 2.07·22-s + 1.48·23-s − 0.747·25-s − 0.348·28-s − 0.995·29-s + 1.39·31-s − 0.0198·32-s + 1.62·34-s − 0.0871·35-s − 1.41·37-s + 0.0641·38-s − 0.875·40-s + 0.487·41-s − 1.34·43-s + 2.39·44-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.90T + 8T^{2} \) |
| 5 | \( 1 - 5.61T + 125T^{2} \) |
| 7 | \( 1 + 3.21T + 343T^{2} \) |
| 11 | \( 1 - 43.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 65.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 3.06T + 6.85e3T^{2} \) |
| 23 | \( 1 - 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 155.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 240.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 317.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 379.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 79.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 571.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 105.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 724.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 302.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 277.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 959.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 410.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 226.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 888.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.905970768161803633099558610243, −8.179485228147733997341744586367, −7.05392453966054500497884281670, −6.71530969862493844428450730749, −5.78237193286065671995165144720, −4.45417954081815548879229172813, −3.13534277347203690354847144154, −1.98956885589002649240866013484, −1.21835728511680407279552464207, 0,
1.21835728511680407279552464207, 1.98956885589002649240866013484, 3.13534277347203690354847144154, 4.45417954081815548879229172813, 5.78237193286065671995165144720, 6.71530969862493844428450730749, 7.05392453966054500497884281670, 8.179485228147733997341744586367, 8.905970768161803633099558610243
