| L(s) = 1 | − 1.08·2-s − 1.29·3-s − 0.825·4-s + 5-s + 1.40·6-s − 5.06·7-s + 3.06·8-s − 1.31·9-s − 1.08·10-s + 11-s + 1.07·12-s + 3.93·13-s + 5.48·14-s − 1.29·15-s − 1.66·16-s − 6.47·17-s + 1.42·18-s + 4.23·19-s − 0.825·20-s + 6.56·21-s − 1.08·22-s − 0.641·23-s − 3.97·24-s + 25-s − 4.26·26-s + 5.60·27-s + 4.18·28-s + ⋯ |
| L(s) = 1 | − 0.766·2-s − 0.748·3-s − 0.412·4-s + 0.447·5-s + 0.573·6-s − 1.91·7-s + 1.08·8-s − 0.439·9-s − 0.342·10-s + 0.301·11-s + 0.309·12-s + 1.09·13-s + 1.46·14-s − 0.334·15-s − 0.416·16-s − 1.57·17-s + 0.336·18-s + 0.971·19-s − 0.184·20-s + 1.43·21-s − 0.231·22-s − 0.133·23-s − 0.810·24-s + 0.200·25-s − 0.835·26-s + 1.07·27-s + 0.790·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.08T + 2T^{2} \) |
| 3 | \( 1 + 1.29T + 3T^{2} \) |
| 7 | \( 1 + 5.06T + 7T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 + 0.641T + 23T^{2} \) |
| 29 | \( 1 + 5.40T + 29T^{2} \) |
| 31 | \( 1 + 8.09T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 6.39T + 41T^{2} \) |
| 43 | \( 1 - 1.89T + 43T^{2} \) |
| 47 | \( 1 - 3.97T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 0.953T + 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 - 6.45T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 79 | \( 1 - 0.693T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.337045851051885696219465255104, −7.17124700072965503711353179836, −6.63837853967211401921151588593, −5.89611069243077549837842051350, −5.39988771960757521701485413626, −4.14980155830327603724527816657, −3.49792898740721544099109891341, −2.34367856750202626307128092309, −0.924968056369230106708478647534, 0,
0.924968056369230106708478647534, 2.34367856750202626307128092309, 3.49792898740721544099109891341, 4.14980155830327603724527816657, 5.39988771960757521701485413626, 5.89611069243077549837842051350, 6.63837853967211401921151588593, 7.17124700072965503711353179836, 8.337045851051885696219465255104
