| L(s) = 1 | + 3-s + 5-s − 2.73·7-s − 2·9-s − 2.73·11-s + 1.73·13-s + 15-s + 5.46·17-s − 2.73·19-s − 2.73·21-s + 23-s + 25-s − 5·27-s + 5.92·29-s + 0.267·31-s − 2.73·33-s − 2.73·35-s + 2.73·37-s + 1.73·39-s + 2.46·41-s − 6.73·43-s − 2·45-s + 5.92·47-s + 0.464·49-s + 5.46·51-s + 0.535·53-s − 2.73·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.03·7-s − 0.666·9-s − 0.823·11-s + 0.480·13-s + 0.258·15-s + 1.32·17-s − 0.626·19-s − 0.596·21-s + 0.208·23-s + 0.200·25-s − 0.962·27-s + 1.10·29-s + 0.0481·31-s − 0.475·33-s − 0.461·35-s + 0.449·37-s + 0.277·39-s + 0.384·41-s − 1.02·43-s − 0.298·45-s + 0.864·47-s + 0.0663·49-s + 0.765·51-s + 0.0736·53-s − 0.368·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 29 | \( 1 - 5.92T + 29T^{2} \) |
| 31 | \( 1 - 0.267T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 - 5.92T + 47T^{2} \) |
| 53 | \( 1 - 0.535T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 7.66T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−7.63818020200118032226837548279, −6.87073591126437149246899614785, −5.97462129512983394624851581116, −5.72861389688663613063425552113, −4.72255505714317392565588587346, −3.71940610262119284610931353134, −2.94867916833247986265263684594, −2.60734540209516801941439535382, −1.32588275026516163104589281609, 0,
1.32588275026516163104589281609, 2.60734540209516801941439535382, 2.94867916833247986265263684594, 3.71940610262119284610931353134, 4.72255505714317392565588587346, 5.72861389688663613063425552113, 5.97462129512983394624851581116, 6.87073591126437149246899614785, 7.63818020200118032226837548279
