| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4466545577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4466545577\) |
| \(L(\frac{5}{2})\) |
|
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
−9.103484428239763757380815784192, −8.255310701113584102055737365140, −7.70210732346850764069569983535, −7.12813488032768843761859641876, −6.04101089281835508892110668458, −4.44924565826895101859961790184, −4.32126864381963202747234487950, −3.07070072674317224582188115469, −2.26689847740700276137625739031, −0.30179698712265435674489547231,
0.30179698712265435674489547231, 2.26689847740700276137625739031, 3.07070072674317224582188115469, 4.32126864381963202747234487950, 4.44924565826895101859961790184, 6.04101089281835508892110668458, 7.12813488032768843761859641876, 7.70210732346850764069569983535, 8.255310701113584102055737365140, 9.103484428239763757380815784192
