L(s) = 1 | + 2.22·3-s − 3.11·5-s + 5.23·7-s + 1.93·9-s − 4.87·11-s + 4.91·13-s − 6.93·15-s − 17-s − 4.18·19-s + 11.6·21-s + 6.48·23-s + 4.72·25-s − 2.35·27-s + 3.77·29-s − 0.169·31-s − 10.8·33-s − 16.3·35-s + 9.64·37-s + 10.9·39-s + 5.58·41-s − 10.5·43-s − 6.04·45-s − 10.9·47-s + 20.4·49-s − 2.22·51-s + 10.6·53-s + 15.2·55-s + ⋯ |
L(s) = 1 | + 1.28·3-s − 1.39·5-s + 1.97·7-s + 0.646·9-s − 1.47·11-s + 1.36·13-s − 1.78·15-s − 0.242·17-s − 0.960·19-s + 2.53·21-s + 1.35·23-s + 0.945·25-s − 0.453·27-s + 0.700·29-s − 0.0304·31-s − 1.88·33-s − 2.76·35-s + 1.58·37-s + 1.74·39-s + 0.872·41-s − 1.60·43-s − 0.901·45-s − 1.60·47-s + 2.91·49-s − 0.311·51-s + 1.46·53-s + 2.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.126610595\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.126610595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 - 6.48T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 + 0.169T + 31T^{2} \) |
| 37 | \( 1 - 9.64T + 37T^{2} \) |
| 41 | \( 1 - 5.58T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 61 | \( 1 + 2.18T + 61T^{2} \) |
| 67 | \( 1 - 6.27T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 8.22T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 2.22T + 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−8.062252091956535947601207445297, −7.56274668145615709218587118006, −6.74539609686571175423537202872, −5.53559907733327178360479168468, −4.76685507954671648193228974629, −4.26426451796187143557364705710, −3.50029958893432369909221014759, −2.70488052361206282876229463876, −1.93929512846404717837335799973, −0.837431887440507573317321153234,
0.837431887440507573317321153234, 1.93929512846404717837335799973, 2.70488052361206282876229463876, 3.50029958893432369909221014759, 4.26426451796187143557364705710, 4.76685507954671648193228974629, 5.53559907733327178360479168468, 6.74539609686571175423537202872, 7.56274668145615709218587118006, 8.062252091956535947601207445297