| L(s) = 1 | + 2·2-s + 4·4-s + 9.19·5-s + 8·8-s + 18.3·10-s − 15.7·13-s + 16·16-s − 33.9·17-s + 36.7·20-s + 59.5·25-s − 31.5·26-s + 16.3·29-s + 32·32-s − 67.9·34-s + 14.2·37-s + 73.5·40-s − 80·41-s + 49·49-s + 119.·50-s − 63.1·52-s − 56·53-s + 32.7·58-s + 114.·61-s + 64·64-s − 145.·65-s − 135.·68-s − 138.·73-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 1.83·5-s + 8-s + 1.83·10-s − 1.21·13-s + 16-s − 1.99·17-s + 1.83·20-s + 2.38·25-s − 1.21·26-s + 0.564·29-s + 32-s − 1.99·34-s + 0.384·37-s + 1.83·40-s − 1.95·41-s + 0.999·49-s + 2.38·50-s − 1.21·52-s − 1.05·53-s + 0.564·58-s + 1.88·61-s + 64-s − 2.23·65-s − 1.99·68-s − 1.89·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.861092315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.861092315\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 9.19T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 15.7T + 169T^{2} \) |
| 17 | \( 1 + 33.9T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 16.3T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 14.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 80T + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 56T + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 114.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 138.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 12.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 130T + 9.40e3T^{2} \) |
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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
−11.48950669104572476838757929399, −10.45500193504466573583405051820, −9.786939398640770324681309439557, −8.687848399865716268265910169773, −7.03510096798253160591415062230, −6.40029985235839871257634173868, −5.35944364862405960347578800248, −4.55313083814263969586736586622, −2.70658746870262121974310267340, −1.90310670384207047914083062323,
1.90310670384207047914083062323, 2.70658746870262121974310267340, 4.55313083814263969586736586622, 5.35944364862405960347578800248, 6.40029985235839871257634173868, 7.03510096798253160591415062230, 8.687848399865716268265910169773, 9.786939398640770324681309439557, 10.45500193504466573583405051820, 11.48950669104572476838757929399
