| L(s) = 1 | + 3·3-s + 20.3·5-s + 7·7-s + 9·9-s − 30.9·11-s + 50.6·13-s + 60.9·15-s − 102.·17-s − 61.2·19-s + 21·21-s + 148.·23-s + 287.·25-s + 27·27-s + 159.·29-s − 121.·31-s − 92.7·33-s + 142.·35-s − 357.·37-s + 151.·39-s + 466.·41-s − 185.·43-s + 182.·45-s − 131.·47-s + 49·49-s − 308.·51-s + 200.·53-s − 627.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.81·5-s + 0.377·7-s + 0.333·9-s − 0.847·11-s + 1.07·13-s + 1.04·15-s − 1.46·17-s − 0.739·19-s + 0.218·21-s + 1.34·23-s + 2.29·25-s + 0.192·27-s + 1.01·29-s − 0.702·31-s − 0.489·33-s + 0.686·35-s − 1.59·37-s + 0.623·39-s + 1.77·41-s − 0.658·43-s + 0.605·45-s − 0.407·47-s + 0.142·49-s − 0.846·51-s + 0.518·53-s − 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.754563640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.754563640\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 20.3T + 125T^{2} \) |
| 11 | \( 1 + 30.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 466.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 185.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 131.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 591.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 70.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 643.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 522.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 576.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 280.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 557.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 65.0T + 9.12e5T^{2} \) |
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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
−12.82968373319003569046052129900, −10.94651193254892048556377888356, −10.39378168409657923298955351062, −9.104576159802684794036175587543, −8.588666547680421905458836578486, −6.93989463786453034573750374663, −5.90653017967864764418028612321, −4.71298521037037433772872947045, −2.76376987435710596838630071607, −1.64662031906487988199545594164,
1.64662031906487988199545594164, 2.76376987435710596838630071607, 4.71298521037037433772872947045, 5.90653017967864764418028612321, 6.93989463786453034573750374663, 8.588666547680421905458836578486, 9.104576159802684794036175587543, 10.39378168409657923298955351062, 10.94651193254892048556377888356, 12.82968373319003569046052129900
