L(s) = 1 | + 2.74·3-s − 3.50·5-s − 7-s + 4.54·9-s − 11-s − 13-s − 9.62·15-s − 0.0357·17-s + 4.00·19-s − 2.74·21-s − 1.32·23-s + 7.28·25-s + 4.22·27-s − 4.14·29-s + 3.90·31-s − 2.74·33-s + 3.50·35-s + 3.16·37-s − 2.74·39-s + 6.72·41-s − 9.33·43-s − 15.9·45-s + 7.15·47-s + 49-s − 0.0982·51-s − 0.562·53-s + 3.50·55-s + ⋯ |
L(s) = 1 | + 1.58·3-s − 1.56·5-s − 0.377·7-s + 1.51·9-s − 0.301·11-s − 0.277·13-s − 2.48·15-s − 0.00868·17-s + 0.919·19-s − 0.599·21-s − 0.276·23-s + 1.45·25-s + 0.813·27-s − 0.770·29-s + 0.701·31-s − 0.478·33-s + 0.592·35-s + 0.520·37-s − 0.439·39-s + 1.05·41-s − 1.42·43-s − 2.37·45-s + 1.04·47-s + 0.142·49-s − 0.0137·51-s − 0.0772·53-s + 0.472·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.289265456\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.289265456\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 17 | \( 1 + 0.0357T + 17T^{2} \) |
| 19 | \( 1 - 4.00T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 + 4.14T + 29T^{2} \) |
| 31 | \( 1 - 3.90T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 + 9.33T + 43T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 + 0.562T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 + 9.28T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 8.34T + 71T^{2} \) |
| 73 | \( 1 + 8.58T + 73T^{2} \) |
| 79 | \( 1 + 6.38T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 3.88T + 97T^{2} \) |
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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
−7.78105029556089868832077512167, −7.50360599812477830657233743427, −6.83785381715676509500949289358, −5.74202061469891294958297166453, −4.67258298160847163967576857949, −4.08314422165281051903272913124, −3.36716684160337613017280972713, −2.96143465593247694865549726786, −2.01843488123534985120075748750, −0.67494517425230898193510373119,
0.67494517425230898193510373119, 2.01843488123534985120075748750, 2.96143465593247694865549726786, 3.36716684160337613017280972713, 4.08314422165281051903272913124, 4.67258298160847163967576857949, 5.74202061469891294958297166453, 6.83785381715676509500949289358, 7.50360599812477830657233743427, 7.78105029556089868832077512167