L(s) = 1 | + 7-s + 3·11-s + 13-s − 6·17-s − 4·19-s + 3·23-s + 3·29-s + 5·31-s − 2·37-s + 3·41-s + 43-s + 9·47-s − 6·49-s + 6·53-s − 3·59-s − 13·61-s + 7·67-s − 12·71-s + 10·73-s + 3·77-s + 11·79-s + 9·83-s + 6·89-s + 91-s − 11·97-s + 15·101-s + 7·103-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 0.904·11-s + 0.277·13-s − 1.45·17-s − 0.917·19-s + 0.625·23-s + 0.557·29-s + 0.898·31-s − 0.328·37-s + 0.468·41-s + 0.152·43-s + 1.31·47-s − 6/7·49-s + 0.824·53-s − 0.390·59-s − 1.66·61-s + 0.855·67-s − 1.42·71-s + 1.17·73-s + 0.341·77-s + 1.23·79-s + 0.987·83-s + 0.635·89-s + 0.104·91-s − 1.11·97-s + 1.49·101-s + 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.155692115\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.155692115\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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
−7.87544534238377520088379376839, −7.00213102822232881705352885471, −6.47771049686538514454754000634, −5.91927084735920075330885890700, −4.78548301360952808660987876110, −4.41542515836709316445480511170, −3.59302669101471453043847906878, −2.57525509504222965285594792602, −1.79537564863292012336161504332, −0.73636935402315542860567052457,
0.73636935402315542860567052457, 1.79537564863292012336161504332, 2.57525509504222965285594792602, 3.59302669101471453043847906878, 4.41542515836709316445480511170, 4.78548301360952808660987876110, 5.91927084735920075330885890700, 6.47771049686538514454754000634, 7.00213102822232881705352885471, 7.87544534238377520088379376839