L(s) = 1 | − 2.60·3-s + 0.822·5-s + 3.53·7-s + 3.79·9-s + 2.95·11-s + 3.38·13-s − 2.14·15-s − 1.72·17-s − 0.518·19-s − 9.21·21-s + 5.52·23-s − 4.32·25-s − 2.06·27-s − 0.947·29-s + 1.67·31-s − 7.69·33-s + 2.90·35-s + 6.48·37-s − 8.83·39-s + 3.34·41-s + 5.45·43-s + 3.11·45-s − 10.5·47-s + 5.49·49-s + 4.49·51-s + 12.3·53-s + 2.42·55-s + ⋯ |
L(s) = 1 | − 1.50·3-s + 0.367·5-s + 1.33·7-s + 1.26·9-s + 0.890·11-s + 0.939·13-s − 0.553·15-s − 0.418·17-s − 0.119·19-s − 2.01·21-s + 1.15·23-s − 0.864·25-s − 0.396·27-s − 0.175·29-s + 0.301·31-s − 1.33·33-s + 0.491·35-s + 1.06·37-s − 1.41·39-s + 0.523·41-s + 0.832·43-s + 0.465·45-s − 1.53·47-s + 0.785·49-s + 0.629·51-s + 1.69·53-s + 0.327·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.873859681\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.873859681\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 + 2.60T + 3T^{2} \) |
| 5 | \( 1 - 0.822T + 5T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 + 0.518T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 + 0.947T + 29T^{2} \) |
| 31 | \( 1 - 1.67T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 - 3.34T + 41T^{2} \) |
| 43 | \( 1 - 5.45T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 0.944T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 5.73T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 7.94T + 97T^{2} \) |
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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
−7.81800194846636819271359174783, −6.75072292075402836413172909246, −6.54594072961005380360499182134, −5.50110977702056980377823971961, −5.36471541899600610355269145613, −4.34953225952344158374578867573, −3.92060792958359218252084939356, −2.43716253358201088359709048174, −1.42448891228927796409036055804, −0.839427798772136795108550308667,
0.839427798772136795108550308667, 1.42448891228927796409036055804, 2.43716253358201088359709048174, 3.92060792958359218252084939356, 4.34953225952344158374578867573, 5.36471541899600610355269145613, 5.50110977702056980377823971961, 6.54594072961005380360499182134, 6.75072292075402836413172909246, 7.81800194846636819271359174783