L(s) = 1 | − 2·3-s + 5-s − 3·7-s + 9-s − 5·11-s + 4·13-s − 2·15-s − 3·17-s + 19-s + 6·21-s + 8·23-s − 4·25-s + 4·27-s + 2·29-s + 4·31-s + 10·33-s − 3·35-s − 10·37-s − 8·39-s + 10·41-s − 43-s + 45-s − 47-s + 2·49-s + 6·51-s + 4·53-s − 5·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 1.50·11-s + 1.10·13-s − 0.516·15-s − 0.727·17-s + 0.229·19-s + 1.30·21-s + 1.66·23-s − 4/5·25-s + 0.769·27-s + 0.371·29-s + 0.718·31-s + 1.74·33-s − 0.507·35-s − 1.64·37-s − 1.28·39-s + 1.56·41-s − 0.152·43-s + 0.149·45-s − 0.145·47-s + 2/7·49-s + 0.840·51-s + 0.549·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7851853327\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7851853327\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−9.888879664431763319657691324377, −9.038018434905016884851105347469, −8.121865683855346250917871835791, −6.89999375237673531400272497166, −6.35171638937504994192666881666, −5.55162300458149510222680140561, −4.93085486739421392203871722376, −3.51647821375355615052320307776, −2.48496537170117614857619865014, −0.67525660115678139253233977173,
0.67525660115678139253233977173, 2.48496537170117614857619865014, 3.51647821375355615052320307776, 4.93085486739421392203871722376, 5.55162300458149510222680140561, 6.35171638937504994192666881666, 6.89999375237673531400272497166, 8.121865683855346250917871835791, 9.038018434905016884851105347469, 9.888879664431763319657691324377