L(s) = 1 | − 3-s + 7-s + 9-s + 13-s + 3·19-s − 21-s + 4·23-s − 27-s − 4·29-s + 7·31-s − 6·37-s − 39-s + 6·41-s − 9·43-s + 6·47-s − 6·49-s + 2·53-s − 3·57-s + 10·59-s + 61-s + 63-s + 3·67-s − 4·69-s + 14·71-s − 10·73-s − 8·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.277·13-s + 0.688·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s − 0.742·29-s + 1.25·31-s − 0.986·37-s − 0.160·39-s + 0.937·41-s − 1.37·43-s + 0.875·47-s − 6/7·49-s + 0.274·53-s − 0.397·57-s + 1.30·59-s + 0.128·61-s + 0.125·63-s + 0.366·67-s − 0.481·69-s + 1.66·71-s − 1.17·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.690337831\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.690337831\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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
−8.324875536974227135043247290169, −7.42232995250695128373017769196, −6.89204965693333458104905257179, −6.04187068356955341408291645451, −5.33203798635269608973392267986, −4.71985856984911950638253443633, −3.81786947612213917397252188735, −2.91455003295055563070651165374, −1.75848962536621981303177711141, −0.76570875824176835921343703982,
0.76570875824176835921343703982, 1.75848962536621981303177711141, 2.91455003295055563070651165374, 3.81786947612213917397252188735, 4.71985856984911950638253443633, 5.33203798635269608973392267986, 6.04187068356955341408291645451, 6.89204965693333458104905257179, 7.42232995250695128373017769196, 8.324875536974227135043247290169