L(s) = 1 | + 5-s − 6·11-s + 2·13-s + 6·17-s + 8·19-s − 3·23-s + 25-s − 3·29-s + 2·31-s + 8·37-s + 3·41-s + 5·43-s − 12·53-s − 6·55-s − 61-s + 2·65-s − 7·67-s − 10·73-s − 4·79-s − 3·83-s + 6·85-s + 3·89-s + 8·95-s − 10·97-s + 3·101-s − 7·103-s + 3·107-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.80·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 0.625·23-s + 1/5·25-s − 0.557·29-s + 0.359·31-s + 1.31·37-s + 0.468·41-s + 0.762·43-s − 1.64·53-s − 0.809·55-s − 0.128·61-s + 0.248·65-s − 0.855·67-s − 1.17·73-s − 0.450·79-s − 0.329·83-s + 0.650·85-s + 0.317·89-s + 0.820·95-s − 1.01·97-s + 0.298·101-s − 0.689·103-s + 0.290·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.207658940\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.207658940\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66998594370642087515569147975, −7.38949688080316772620952760749, −6.11792192194587947850632781762, −5.68524275880525065137524693514, −5.17808413628322194122986571322, −4.31112151530658401932981549524, −3.17055185618460234714963623953, −2.85660724488780619851728556935, −1.70685655145218414627559973764, −0.73691252128085280602083961922,
0.73691252128085280602083961922, 1.70685655145218414627559973764, 2.85660724488780619851728556935, 3.17055185618460234714963623953, 4.31112151530658401932981549524, 5.17808413628322194122986571322, 5.68524275880525065137524693514, 6.11792192194587947850632781762, 7.38949688080316772620952760749, 7.66998594370642087515569147975