| L(s) = 1 | − 85.7·3-s + 1.78e3·5-s + 751.·7-s − 1.23e4·9-s − 5.85e4·11-s − 2.85e4·13-s − 1.53e5·15-s + 2.34e5·17-s + 1.03e5·19-s − 6.44e4·21-s − 1.08e6·23-s + 1.24e6·25-s + 2.74e6·27-s + 3.17e6·29-s + 6.17e6·31-s + 5.02e6·33-s + 1.34e6·35-s − 1.56e7·37-s + 2.44e6·39-s − 2.01e7·41-s + 2.69e7·43-s − 2.20e7·45-s + 3.03e7·47-s − 3.97e7·49-s − 2.01e7·51-s + 6.56e7·53-s − 1.04e8·55-s + ⋯ |
| L(s) = 1 | − 0.611·3-s + 1.27·5-s + 0.118·7-s − 0.626·9-s − 1.20·11-s − 0.277·13-s − 0.781·15-s + 0.681·17-s + 0.181·19-s − 0.0723·21-s − 0.810·23-s + 0.635·25-s + 0.994·27-s + 0.833·29-s + 1.20·31-s + 0.736·33-s + 0.151·35-s − 1.36·37-s + 0.169·39-s − 1.11·41-s + 1.20·43-s − 0.801·45-s + 0.906·47-s − 0.985·49-s − 0.416·51-s + 1.14·53-s − 1.54·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.733359088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.733359088\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 3 | \( 1 + 85.7T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.78e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 751.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.85e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 2.34e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.03e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.08e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.17e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.17e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.56e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.01e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.69e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.56e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.12e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.44e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.29e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.77e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.05e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.62e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.15e8T + 7.60e17T^{2} \) |
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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
−10.44678996870530014138975144886, −10.11631937151194731078902644929, −8.826764717393155714352547375814, −7.75555946496839817723145013954, −6.37323908222377420574557372629, −5.60503785771485445661039161250, −4.88905290085430168690570227121, −3.00758231802271605251057444847, −2.02116934769278117157710554701, −0.63077238981532028815178317176,
0.63077238981532028815178317176, 2.02116934769278117157710554701, 3.00758231802271605251057444847, 4.88905290085430168690570227121, 5.60503785771485445661039161250, 6.37323908222377420574557372629, 7.75555946496839817723145013954, 8.826764717393155714352547375814, 10.11631937151194731078902644929, 10.44678996870530014138975144886
