L(s) = 1 | + 2.56·2-s + 4.56·4-s − 3.56·5-s + 6.56·8-s − 9.12·10-s − 1.56·11-s + 0.438·13-s + 7.68·16-s − 17-s − 4.68·19-s − 16.2·20-s − 4·22-s + 2.43·23-s + 7.68·25-s + 1.12·26-s + 8.24·29-s + 3.12·31-s + 6.56·32-s − 2.56·34-s − 5.12·37-s − 11.9·38-s − 23.3·40-s + 3.56·41-s + 4.68·43-s − 7.12·44-s + 6.24·46-s + 11.1·47-s + ⋯ |
L(s) = 1 | + 1.81·2-s + 2.28·4-s − 1.59·5-s + 2.31·8-s − 2.88·10-s − 0.470·11-s + 0.121·13-s + 1.92·16-s − 0.242·17-s − 1.07·19-s − 3.63·20-s − 0.852·22-s + 0.508·23-s + 1.53·25-s + 0.220·26-s + 1.53·29-s + 0.560·31-s + 1.15·32-s − 0.439·34-s − 0.842·37-s − 1.94·38-s − 3.69·40-s + 0.556·41-s + 0.714·43-s − 1.07·44-s + 0.920·46-s + 1.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.333534695\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.333534695\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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
−12.81153383843558361801241281824, −12.26798724154686299614565601806, −11.33738186083091183668737219454, −10.62539255162187764834405087385, −8.452248548809288367614664414456, −7.37400369714000213035983918020, −6.35018764485142958760593183486, −4.85073210575943155801207233947, −4.06209328209427631305994086121, −2.86226885331132817253176394642,
2.86226885331132817253176394642, 4.06209328209427631305994086121, 4.85073210575943155801207233947, 6.35018764485142958760593183486, 7.37400369714000213035983918020, 8.452248548809288367614664414456, 10.62539255162187764834405087385, 11.33738186083091183668737219454, 12.26798724154686299614565601806, 12.81153383843558361801241281824