| L(s) = 1 | − 3-s + 2.52·5-s + 0.307·7-s + 9-s + 3.21·11-s − 3.88·13-s − 2.52·15-s − 4.06·17-s + 1.71·19-s − 0.307·21-s − 2.23·23-s + 1.35·25-s − 27-s + 2.71·29-s − 9.46·31-s − 3.21·33-s + 0.774·35-s + 8.47·37-s + 3.88·39-s + 1.17·41-s + 0.594·43-s + 2.52·45-s − 6.00·47-s − 6.90·49-s + 4.06·51-s − 5.37·53-s + 8.10·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.12·5-s + 0.116·7-s + 0.333·9-s + 0.968·11-s − 1.07·13-s − 0.651·15-s − 0.986·17-s + 0.394·19-s − 0.0670·21-s − 0.466·23-s + 0.271·25-s − 0.192·27-s + 0.504·29-s − 1.70·31-s − 0.559·33-s + 0.130·35-s + 1.39·37-s + 0.622·39-s + 0.184·41-s + 0.0906·43-s + 0.375·45-s − 0.875·47-s − 0.986·49-s + 0.569·51-s − 0.738·53-s + 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 - 0.307T + 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 - 2.71T + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 - 0.594T + 43T^{2} \) |
| 47 | \( 1 + 6.00T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 + 4.70T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 - 3.54T + 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 + 8.22T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 8.83T + 83T^{2} \) |
| 89 | \( 1 + 8.38T + 89T^{2} \) |
| 97 | \( 1 + 6.81T + 97T^{2} \) |
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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
−7.27126863118120153522741070413, −6.72913343057205181416561104875, −6.04896677019405026031242615254, −5.53191107221541267689432737568, −4.70648638882064593060914934113, −4.13477572683650212702340771612, −2.97068747114535280899093553111, −2.05265910251596374242728306217, −1.40410427894647138274563020493, 0,
1.40410427894647138274563020493, 2.05265910251596374242728306217, 2.97068747114535280899093553111, 4.13477572683650212702340771612, 4.70648638882064593060914934113, 5.53191107221541267689432737568, 6.04896677019405026031242615254, 6.72913343057205181416561104875, 7.27126863118120153522741070413
