L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 4·11-s − 6·13-s − 15-s − 6·17-s + 19-s − 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s + 8·31-s − 4·33-s + 4·35-s + 2·37-s + 6·39-s + 10·41-s + 8·43-s + 45-s − 12·47-s + 9·49-s + 6·51-s + 2·53-s + 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.258·15-s − 1.45·17-s + 0.229·19-s − 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s + 0.960·39-s + 1.56·41-s + 1.21·43-s + 0.149·45-s − 1.75·47-s + 9/7·49-s + 0.840·51-s + 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.079995205\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.079995205\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−8.277627112389515787405107563696, −7.59390343028989364593197256105, −6.76871884253750954653435970463, −6.22486279503382217386656381472, −5.26176815262399212025932520324, −4.54902018642936320716355836435, −4.26336581468753684039224074587, −2.59353318098753771955118975132, −1.91300121549708485528381524911, −0.855628027457763073416109508882,
0.855628027457763073416109508882, 1.91300121549708485528381524911, 2.59353318098753771955118975132, 4.26336581468753684039224074587, 4.54902018642936320716355836435, 5.26176815262399212025932520324, 6.22486279503382217386656381472, 6.76871884253750954653435970463, 7.59390343028989364593197256105, 8.277627112389515787405107563696