L(s) = 1 | − 2.78·3-s − 5-s − 7-s + 4.75·9-s − 4.82·11-s + 2.97·13-s + 2.78·15-s + 0.905·17-s − 2.18·19-s + 2.78·21-s − 8.65·23-s + 25-s − 4.90·27-s + 1.92·29-s − 1.24·31-s + 13.4·33-s + 35-s − 1.30·37-s − 8.28·39-s − 1.94·41-s − 43-s − 4.75·45-s − 10.4·47-s + 49-s − 2.52·51-s + 10.1·53-s + 4.82·55-s + ⋯ |
L(s) = 1 | − 1.60·3-s − 0.447·5-s − 0.377·7-s + 1.58·9-s − 1.45·11-s + 0.824·13-s + 0.719·15-s + 0.219·17-s − 0.502·19-s + 0.607·21-s − 1.80·23-s + 0.200·25-s − 0.943·27-s + 0.357·29-s − 0.224·31-s + 2.33·33-s + 0.169·35-s − 0.213·37-s − 1.32·39-s − 0.303·41-s − 0.152·43-s − 0.709·45-s − 1.52·47-s + 0.142·49-s − 0.353·51-s + 1.39·53-s + 0.650·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2898436008\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2898436008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 - 0.905T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 + 8.65T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 + 1.94T + 41T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.71T + 59T^{2} \) |
| 61 | \( 1 + 4.49T + 61T^{2} \) |
| 67 | \( 1 + 8.65T + 67T^{2} \) |
| 71 | \( 1 + 4.78T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 8.82T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 3.11T + 97T^{2} \) |
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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
−8.047782501976231256967042389860, −7.21986636813258331547246511668, −6.52010092147013128760823889015, −5.84693183229735719524635012682, −5.41410037725045826901037127210, −4.54795611598138665185100378785, −3.87606672354750265133146269311, −2.82191398131798642932290092936, −1.59654640506344339380222894428, −0.30707459570146219777572559920,
0.30707459570146219777572559920, 1.59654640506344339380222894428, 2.82191398131798642932290092936, 3.87606672354750265133146269311, 4.54795611598138665185100378785, 5.41410037725045826901037127210, 5.84693183229735719524635012682, 6.52010092147013128760823889015, 7.21986636813258331547246511668, 8.047782501976231256967042389860