L(s) = 1 | + (1.41 − 0.0765i)2-s + 2.21i·3-s + (1.98 − 0.216i)4-s − 5-s + (0.169 + 3.13i)6-s + (−1.20 + 2.35i)7-s + (2.79 − 0.457i)8-s − 1.92·9-s + (−1.41 + 0.0765i)10-s − 3.88·11-s + (0.479 + 4.41i)12-s + 5.67·13-s + (−1.52 + 3.41i)14-s − 2.21i·15-s + (3.90 − 0.859i)16-s − 5.63i·17-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0541i)2-s + 1.28i·3-s + (0.994 − 0.108i)4-s − 0.447·5-s + (0.0693 + 1.27i)6-s + (−0.457 + 0.889i)7-s + (0.986 − 0.161i)8-s − 0.641·9-s + (−0.446 + 0.0242i)10-s − 1.17·11-s + (0.138 + 1.27i)12-s + 1.57·13-s + (−0.408 + 0.912i)14-s − 0.572i·15-s + (0.976 − 0.214i)16-s − 1.36i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.70794 + 1.24338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70794 + 1.24338i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 + 0.0765i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (1.20 - 2.35i)T \) |
good | 3 | \( 1 - 2.21iT - 3T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 13 | \( 1 - 5.67T + 13T^{2} \) |
| 17 | \( 1 + 5.63iT - 17T^{2} \) |
| 19 | \( 1 - 1.31iT - 19T^{2} \) |
| 23 | \( 1 + 7.37iT - 23T^{2} \) |
| 29 | \( 1 - 9.07iT - 29T^{2} \) |
| 31 | \( 1 - 2.23T + 31T^{2} \) |
| 37 | \( 1 + 6.98iT - 37T^{2} \) |
| 41 | \( 1 + 7.47iT - 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 0.567T + 47T^{2} \) |
| 53 | \( 1 + 0.100iT - 53T^{2} \) |
| 59 | \( 1 + 2.93iT - 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 5.54T + 67T^{2} \) |
| 71 | \( 1 + 2.42iT - 71T^{2} \) |
| 73 | \( 1 - 6.08iT - 73T^{2} \) |
| 79 | \( 1 - 2.83iT - 79T^{2} \) |
| 83 | \( 1 - 2.52iT - 83T^{2} \) |
| 89 | \( 1 - 10.9iT - 89T^{2} \) |
| 97 | \( 1 - 9.93iT - 97T^{2} \) |
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
−12.13522249478705997736476255156, −10.91224421399832182272839607234, −10.60281331151720513058298265897, −9.308687275480353289239841919315, −8.310742448978218953497027099484, −6.86857991890527777469416110106, −5.61966627219753628309851546585, −4.87076313859400790453364508535, −3.70131725686429513555442976268, −2.76585507219590495353782929059,
1.45202382180931990654546450156, 3.15693719258318755703834038605, 4.26727421595251541699378593184, 5.91994472563318274823727154320, 6.56938787442048167643405352216, 7.71469776937117609018126116536, 8.101795154511665379896566825321, 10.15843771477736102317812245566, 11.06515797111316269609525848188, 11.86683587601399997431071337935