| L(s) = 1 | + 2·2-s + 4·4-s + 9.48·5-s − 3.48·7-s + 8·8-s + 18.9·10-s − 21.9·11-s − 6.97·14-s + 16·16-s + 37.9·20-s − 43.9·22-s + 64.9·25-s − 13.9·28-s − 50·29-s + 23.4·31-s + 32·32-s − 33.0·35-s + 75.8·40-s − 87.8·44-s − 36.8·49-s + 129.·50-s − 89.4·53-s − 208.·55-s − 27.8·56-s − 100·58-s + 10·59-s + 46.8·62-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 1.89·5-s − 0.497·7-s + 8-s + 1.89·10-s − 1.99·11-s − 0.497·14-s + 16-s + 1.89·20-s − 1.99·22-s + 2.59·25-s − 0.497·28-s − 1.72·29-s + 0.755·31-s + 32-s − 0.944·35-s + 1.89·40-s − 1.99·44-s − 0.752·49-s + 2.59·50-s − 1.68·53-s − 3.78·55-s − 0.497·56-s − 1.72·58-s + 0.169·59-s + 0.755·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.323276912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.323276912\) |
| \(L(2)\) |
|
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 | \( 1 - 9.48T + 25T^{2} \) |
| 7 | \( 1 + 3.48T + 49T^{2} \) |
| 11 | \( 1 + 21.9T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 50T + 841T^{2} \) |
| 31 | \( 1 - 23.4T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 89.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 10T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 93.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58T + 6.24e3T^{2} \) |
| 83 | \( 1 - 151.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 61.0T + 9.40e3T^{2} \) |
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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
−12.62151192715286089363614937032, −11.00066963430168686405502360083, −10.27713823805249923090290550963, −9.484174955687653977700448194286, −7.85879263318750141785987509295, −6.57275785743757410232237913225, −5.67799324969658571577223763054, −4.99919492001644686267661053982, −3.00884831610442378889702866765, −2.03561600933202312448902025052,
2.03561600933202312448902025052, 3.00884831610442378889702866765, 4.99919492001644686267661053982, 5.67799324969658571577223763054, 6.57275785743757410232237913225, 7.85879263318750141785987509295, 9.484174955687653977700448194286, 10.27713823805249923090290550963, 11.00066963430168686405502360083, 12.62151192715286089363614937032
