L(s) = 1 | + 2-s + 4-s + 1.53·5-s − 3.69·7-s + 8-s + 1.53·10-s − 0.317·11-s + 4.82·13-s − 3.69·14-s + 16-s + 4.82·19-s + 1.53·20-s − 0.317·22-s − 7.39·23-s − 2.65·25-s + 4.82·26-s − 3.69·28-s + 3.69·29-s + 1.53·31-s + 32-s − 5.65·35-s + 5.22·37-s + 4.82·38-s + 1.53·40-s + 6.17·41-s + 10.2·43-s − 0.317·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.684·5-s − 1.39·7-s + 0.353·8-s + 0.484·10-s − 0.0955·11-s + 1.33·13-s − 0.987·14-s + 0.250·16-s + 1.10·19-s + 0.342·20-s − 0.0675·22-s − 1.54·23-s − 0.531·25-s + 0.946·26-s − 0.698·28-s + 0.686·29-s + 0.274·31-s + 0.176·32-s − 0.956·35-s + 0.859·37-s + 0.783·38-s + 0.242·40-s + 0.964·41-s + 1.56·43-s − 0.0477·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.274267628\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.274267628\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 0.317T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 + 7.39T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 4.82T + 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 - 4.58T + 67T^{2} \) |
| 71 | \( 1 + 5.22T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 2.16T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 2.29T + 97T^{2} \) |
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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
−8.061990543044908971288961131514, −7.39879270104180274316964495859, −6.29960172978230036723477075864, −6.15663761147228996973192083951, −5.56234853584644884557829376219, −4.39622047500314879058974683586, −3.67457864909137619658897424586, −2.99207603301817452623315204601, −2.11408794640679895214660600150, −0.890567154618777163368984033586,
0.890567154618777163368984033586, 2.11408794640679895214660600150, 2.99207603301817452623315204601, 3.67457864909137619658897424586, 4.39622047500314879058974683586, 5.56234853584644884557829376219, 6.15663761147228996973192083951, 6.29960172978230036723477075864, 7.39879270104180274316964495859, 8.061990543044908971288961131514