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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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
−8.898388983103280563532982969003, −7.75314975862125950084012002168, −7.32790981849486330907346223526, −6.58572205966979815593786534839, −5.27507144677204813777710233267, −4.44228088863160655126473648652, −3.91109382706976883806622938279, −2.63765087168340060138996362987, −1.01237081630015100837115822755, 0,
1.01237081630015100837115822755, 2.63765087168340060138996362987, 3.91109382706976883806622938279, 4.44228088863160655126473648652, 5.27507144677204813777710233267, 6.58572205966979815593786534839, 7.32790981849486330907346223526, 7.75314975862125950084012002168, 8.898388983103280563532982969003