L(s) = 1 | − 1.56·3-s + 0.844·5-s + 5.06·7-s − 0.561·9-s + 2.84·11-s − 2.90·13-s − 1.31·15-s + 7.72·17-s + 19-s − 7.91·21-s − 3.91·23-s − 4.28·25-s + 5.56·27-s − 2.90·29-s + 7.00·31-s − 4.44·33-s + 4.27·35-s − 9.00·37-s + 4.53·39-s − 6.81·41-s + 5.96·43-s − 0.474·45-s + 1.04·47-s + 18.6·49-s − 12.0·51-s + 9.03·53-s + 2.40·55-s + ⋯ |
L(s) = 1 | − 0.901·3-s + 0.377·5-s + 1.91·7-s − 0.187·9-s + 0.857·11-s − 0.804·13-s − 0.340·15-s + 1.87·17-s + 0.229·19-s − 1.72·21-s − 0.815·23-s − 0.857·25-s + 1.07·27-s − 0.538·29-s + 1.25·31-s − 0.773·33-s + 0.723·35-s − 1.48·37-s + 0.725·39-s − 1.06·41-s + 0.910·43-s − 0.0707·45-s + 0.151·47-s + 2.66·49-s − 1.68·51-s + 1.24·53-s + 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.625805220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.625805220\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 0.844T + 5T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 - 7.72T + 17T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 + 9.00T + 37T^{2} \) |
| 41 | \( 1 + 6.81T + 41T^{2} \) |
| 43 | \( 1 - 5.96T + 43T^{2} \) |
| 47 | \( 1 - 1.04T + 47T^{2} \) |
| 53 | \( 1 - 9.03T + 53T^{2} \) |
| 59 | \( 1 - 0.749T + 59T^{2} \) |
| 61 | \( 1 + 2.27T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 + 3.88T + 79T^{2} \) |
| 83 | \( 1 - 8.44T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−9.984717418486069826058117385089, −8.850747929723086901585969437800, −8.009574157247974528123517401790, −7.36587527251920445977263358886, −6.17197129387899924194818303283, −5.38934180538649206058523583571, −4.92573797800613736282758985059, −3.75250334344436521117111165906, −2.13620304908082963292739725866, −1.08078416561479399268482644582,
1.08078416561479399268482644582, 2.13620304908082963292739725866, 3.75250334344436521117111165906, 4.92573797800613736282758985059, 5.38934180538649206058523583571, 6.17197129387899924194818303283, 7.36587527251920445977263358886, 8.009574157247974528123517401790, 8.850747929723086901585969437800, 9.984717418486069826058117385089