| L(s) = 1 | − 2.71·2-s − 3-s + 5.38·4-s + 2.54·5-s + 2.71·6-s + 4.22·7-s − 9.20·8-s + 9-s − 6.92·10-s − 2.27·11-s − 5.38·12-s − 4.46·13-s − 11.4·14-s − 2.54·15-s + 14.2·16-s − 17-s − 2.71·18-s + 3.01·19-s + 13.7·20-s − 4.22·21-s + 6.18·22-s + 4.52·23-s + 9.20·24-s + 1.49·25-s + 12.1·26-s − 27-s + 22.7·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 0.577·3-s + 2.69·4-s + 1.13·5-s + 1.10·6-s + 1.59·7-s − 3.25·8-s + 0.333·9-s − 2.19·10-s − 0.686·11-s − 1.55·12-s − 1.23·13-s − 3.06·14-s − 0.657·15-s + 3.56·16-s − 0.242·17-s − 0.640·18-s + 0.691·19-s + 3.06·20-s − 0.921·21-s + 1.31·22-s + 0.944·23-s + 1.87·24-s + 0.298·25-s + 2.37·26-s − 0.192·27-s + 4.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9898709523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9898709523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 19 | \( 1 - 3.01T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 + 0.921T + 29T^{2} \) |
| 31 | \( 1 - 9.72T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 2.71T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 - 8.77T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 - 7.24T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 8.60T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 4.75T + 83T^{2} \) |
| 89 | \( 1 + 6.81T + 89T^{2} \) |
| 97 | \( 1 + 2.48T + 97T^{2} \) |
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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
−7.83444196643228190410553992375, −7.35029534371811906542933087693, −6.79596963070695910272000728224, −5.82017012011616494367133303890, −5.32700170778315396249640288080, −4.62291960876506547856043988450, −2.85506841172964644474658699296, −2.21383800586247844515139331353, −1.55571840565772712249623822324, −0.70716947090304140515664154554,
0.70716947090304140515664154554, 1.55571840565772712249623822324, 2.21383800586247844515139331353, 2.85506841172964644474658699296, 4.62291960876506547856043988450, 5.32700170778315396249640288080, 5.82017012011616494367133303890, 6.79596963070695910272000728224, 7.35029534371811906542933087693, 7.83444196643228190410553992375
