| L(s) = 1 | + 1.73·2-s + 0.999·4-s − 3.46·7-s − 1.73·8-s + 3.46·11-s − 5.99·14-s − 5·16-s − 6·17-s − 3.46·19-s + 5.99·22-s − 5·25-s − 3.46·28-s − 6·29-s + 3.46·31-s − 5.19·32-s − 10.3·34-s + 6.92·37-s − 5.99·38-s − 6.92·41-s + 4·43-s + 3.46·44-s − 3.46·47-s + 4.99·49-s − 8.66·50-s − 6·53-s + 6.00·56-s − 10.3·58-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s − 1.30·7-s − 0.612·8-s + 1.04·11-s − 1.60·14-s − 1.25·16-s − 1.45·17-s − 0.794·19-s + 1.27·22-s − 25-s − 0.654·28-s − 1.11·29-s + 0.622·31-s − 0.918·32-s − 1.78·34-s + 1.13·37-s − 0.973·38-s − 1.08·41-s + 0.609·43-s + 0.522·44-s − 0.505·47-s + 0.714·49-s − 1.22·50-s − 0.824·53-s + 0.801·56-s − 1.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
| 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
−9.315039277666539569677827769091, −8.348056378507868857374601056029, −6.99861178288976733507776813503, −6.35312880114774201187140407245, −5.93944329937620013237831410201, −4.64589706219920012117607184766, −3.99881600976933982522072013808, −3.24276504929977198909387745568, −2.13982618907773264056735201748, 0,
2.13982618907773264056735201748, 3.24276504929977198909387745568, 3.99881600976933982522072013808, 4.64589706219920012117607184766, 5.93944329937620013237831410201, 6.35312880114774201187140407245, 6.99861178288976733507776813503, 8.348056378507868857374601056029, 9.315039277666539569677827769091
