| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4·11-s + 2·13-s − 14-s + 16-s + 17-s + 4·19-s − 4·22-s + 8·23-s + 2·26-s − 28-s − 6·29-s + 32-s + 34-s + 2·37-s + 4·38-s − 10·41-s + 4·43-s − 4·44-s + 8·46-s + 49-s + 2·52-s + 6·53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.20·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.852·22-s + 1.66·23-s + 0.392·26-s − 0.188·28-s − 1.11·29-s + 0.176·32-s + 0.171·34-s + 0.328·37-s + 0.648·38-s − 1.56·41-s + 0.609·43-s − 0.603·44-s + 1.17·46-s + 1/7·49-s + 0.277·52-s + 0.824·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.634800373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.634800373\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.33467579887119, −13.89453966323142, −13.28334220505923, −13.00328779568950, −12.64928331092936, −11.91399393517697, −11.29482022295949, −11.07755437903584, −10.32333206321893, −9.963968053061162, −9.258831068584352, −8.734330677180255, −8.011407384165695, −7.572895061673066, −6.930985074576201, −6.553113776397495, −5.655851395540180, −5.287158810336325, −4.972130937733457, −3.936544876924875, −3.540090755288284, −2.859043606956638, −2.368883295761019, −1.404641706912393, −0.6009670475093881,
0.6009670475093881, 1.404641706912393, 2.368883295761019, 2.859043606956638, 3.540090755288284, 3.936544876924875, 4.972130937733457, 5.287158810336325, 5.655851395540180, 6.553113776397495, 6.930985074576201, 7.572895061673066, 8.011407384165695, 8.734330677180255, 9.258831068584352, 9.963968053061162, 10.32333206321893, 11.07755437903584, 11.29482022295949, 11.91399393517697, 12.64928331092936, 13.00328779568950, 13.28334220505923, 13.89453966323142, 14.33467579887119
