| L(s) = 1 | − 2-s + 3-s + 4-s + 3.46·5-s − 6-s − 8-s + 9-s − 3.46·10-s − 5.46·11-s + 12-s + 3.46·13-s + 3.46·15-s + 16-s − 7.46·17-s − 18-s − 19-s + 3.46·20-s + 5.46·22-s − 5.46·23-s − 24-s + 6.99·25-s − 3.46·26-s + 27-s − 6·29-s − 3.46·30-s − 6.92·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.54·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.09·10-s − 1.64·11-s + 0.288·12-s + 0.960·13-s + 0.894·15-s + 0.250·16-s − 1.81·17-s − 0.235·18-s − 0.229·19-s + 0.774·20-s + 1.16·22-s − 1.13·23-s − 0.204·24-s + 1.39·25-s − 0.679·26-s + 0.192·27-s − 1.11·29-s − 0.632·30-s − 1.24·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 9.46T + 67T^{2} \) |
| 71 | \( 1 - 2.92T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 4.53T + 97T^{2} \) |
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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
−7.992623530692950454339934098052, −7.16433466454858379842460664965, −6.33209536284402095642074737346, −5.83520585735170778549203290538, −5.04440351078505680859734675380, −3.99046128636283841912109211178, −2.84301183057104393501007275263, −2.16733463971007221010120316165, −1.66890000642658293643010932139, 0,
1.66890000642658293643010932139, 2.16733463971007221010120316165, 2.84301183057104393501007275263, 3.99046128636283841912109211178, 5.04440351078505680859734675380, 5.83520585735170778549203290538, 6.33209536284402095642074737346, 7.16433466454858379842460664965, 7.992623530692950454339934098052
