| L(s) = 1 | + 2-s + 4-s + 5-s + 2.69·7-s + 8-s + 10-s + 3.22·11-s + 1.66·13-s + 2.69·14-s + 16-s + 0.666·17-s − 5.66·19-s + 20-s + 3.22·22-s + 3.67·23-s + 25-s + 1.66·26-s + 2.69·28-s − 9.13·29-s + 5.71·31-s + 32-s + 0.666·34-s + 2.69·35-s + 5.79·37-s − 5.66·38-s + 40-s − 8.92·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.01·7-s + 0.353·8-s + 0.316·10-s + 0.973·11-s + 0.462·13-s + 0.719·14-s + 0.250·16-s + 0.161·17-s − 1.29·19-s + 0.223·20-s + 0.688·22-s + 0.766·23-s + 0.200·25-s + 0.326·26-s + 0.508·28-s − 1.69·29-s + 1.02·31-s + 0.176·32-s + 0.114·34-s + 0.454·35-s + 0.952·37-s − 0.918·38-s + 0.158·40-s − 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.772381794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.772381794\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 - 0.666T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 - 3.67T + 23T^{2} \) |
| 29 | \( 1 + 9.13T + 29T^{2} \) |
| 31 | \( 1 - 5.71T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 6.74T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 0.318T + 73T^{2} \) |
| 79 | \( 1 + 1.09T + 79T^{2} \) |
| 83 | \( 1 - 7.18T + 83T^{2} \) |
| 97 | \( 1 - 9.35T + 97T^{2} \) |
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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
−7.67288176319572183698420746824, −7.08539839576963744445808541156, −6.24399704988402586327762786560, −5.81094508150693334044229404763, −4.97550689808420282153460353742, −4.27771211574907971860790900454, −3.74645528755325019232001868708, −2.62202890379762495104005330835, −1.85722891402777478020609166049, −1.05244374222143634470201423713,
1.05244374222143634470201423713, 1.85722891402777478020609166049, 2.62202890379762495104005330835, 3.74645528755325019232001868708, 4.27771211574907971860790900454, 4.97550689808420282153460353742, 5.81094508150693334044229404763, 6.24399704988402586327762786560, 7.08539839576963744445808541156, 7.67288176319572183698420746824
