| L(s) = 1 | − 2·4-s + 5-s + 7-s − 2·13-s + 4·16-s + 6·17-s + 7·19-s − 2·20-s + 6·23-s + 25-s − 2·28-s − 31-s + 35-s − 7·37-s − 6·41-s − 8·43-s − 6·49-s + 4·52-s + 6·53-s + 12·59-s + 61-s − 8·64-s − 2·65-s − 7·67-s − 12·68-s − 6·71-s + 13·73-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.377·7-s − 0.554·13-s + 16-s + 1.45·17-s + 1.60·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.377·28-s − 0.179·31-s + 0.169·35-s − 1.15·37-s − 0.937·41-s − 1.21·43-s − 6/7·49-s + 0.554·52-s + 0.824·53-s + 1.56·59-s + 0.128·61-s − 64-s − 0.248·65-s − 0.855·67-s − 1.45·68-s − 0.712·71-s + 1.52·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.879789637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.879789637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−8.251682372974885900517389334109, −7.48183805367345486398243608885, −6.89941504053782421909068297312, −5.67308394770304249537577495980, −5.22472911811469780563639893978, −4.77701839527008824350664999436, −3.53808548088749155534040664038, −3.08864933269452777226840388808, −1.66877382617649710238137662445, −0.792360371301422076792621127014,
0.792360371301422076792621127014, 1.66877382617649710238137662445, 3.08864933269452777226840388808, 3.53808548088749155534040664038, 4.77701839527008824350664999436, 5.22472911811469780563639893978, 5.67308394770304249537577495980, 6.89941504053782421909068297312, 7.48183805367345486398243608885, 8.251682372974885900517389334109
