| L(s) = 1 | + 2.76·2-s + 2.46·3-s + 5.65·4-s − 5-s + 6.81·6-s − 7-s + 10.1·8-s + 3.06·9-s − 2.76·10-s − 2.78·11-s + 13.9·12-s + 3.88·13-s − 2.76·14-s − 2.46·15-s + 16.6·16-s − 4.28·17-s + 8.49·18-s + 2.70·19-s − 5.65·20-s − 2.46·21-s − 7.70·22-s + 6.92·23-s + 24.9·24-s + 25-s + 10.7·26-s + 0.171·27-s − 5.65·28-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 1.42·3-s + 2.82·4-s − 0.447·5-s + 2.78·6-s − 0.377·7-s + 3.57·8-s + 1.02·9-s − 0.874·10-s − 0.839·11-s + 4.02·12-s + 1.07·13-s − 0.739·14-s − 0.636·15-s + 4.16·16-s − 1.03·17-s + 2.00·18-s + 0.620·19-s − 1.26·20-s − 0.537·21-s − 1.64·22-s + 1.44·23-s + 5.08·24-s + 0.200·25-s + 2.10·26-s + 0.0330·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.35136419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.35136419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 3 | \( 1 - 2.46T + 3T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 13 | \( 1 - 3.88T + 13T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 8.72T + 29T^{2} \) |
| 31 | \( 1 - 9.05T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 4.06T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 8.15T + 71T^{2} \) |
| 73 | \( 1 + 7.93T + 73T^{2} \) |
| 79 | \( 1 + 9.55T + 79T^{2} \) |
| 83 | \( 1 + 2.97T + 83T^{2} \) |
| 89 | \( 1 - 8.83T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.54759486617683824709977416747, −7.14297945394656481151093489982, −6.34827861920610398372323999495, −5.62416648080561514435440003164, −4.81445852588694220601473647881, −4.10548376976878702656898964872, −3.53482613948130331078008935544, −2.86663996693970523077416839873, −2.47254514585587480230826240114, −1.35683279169193078115127865763,
1.35683279169193078115127865763, 2.47254514585587480230826240114, 2.86663996693970523077416839873, 3.53482613948130331078008935544, 4.10548376976878702656898964872, 4.81445852588694220601473647881, 5.62416648080561514435440003164, 6.34827861920610398372323999495, 7.14297945394656481151093489982, 7.54759486617683824709977416747
