| L(s) = 1 | − 3·3-s + 9.79·5-s + 9.79·7-s + 9·9-s − 10·11-s − 29.3·15-s − 29.3·21-s + 70.9·25-s − 27·27-s − 29.3·29-s + 48.9·31-s + 30·33-s + 95.9·35-s + 88.1·45-s + 46.9·49-s + 48.9·53-s − 97.9·55-s + 10·59-s + 88.1·63-s − 50·73-s − 212.·75-s − 97.9·77-s − 146.·79-s + 81·81-s + 134·83-s + 88.1·87-s − 146.·93-s + ⋯ |
| L(s) = 1 | − 3-s + 1.95·5-s + 1.39·7-s + 9-s − 0.909·11-s − 1.95·15-s − 1.39·21-s + 2.83·25-s − 27-s − 1.01·29-s + 1.58·31-s + 0.909·33-s + 2.74·35-s + 1.95·45-s + 0.959·49-s + 0.924·53-s − 1.78·55-s + 0.169·59-s + 1.39·63-s − 0.684·73-s − 2.83·75-s − 1.27·77-s − 1.86·79-s + 81-s + 1.61·83-s + 1.01·87-s − 1.58·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.972811536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.972811536\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| good | 5 | \( 1 - 9.79T + 25T^{2} \) |
| 7 | \( 1 - 9.79T + 49T^{2} \) |
| 11 | \( 1 + 10T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 29.3T + 841T^{2} \) |
| 31 | \( 1 - 48.9T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 10T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 50T + 5.32e3T^{2} \) |
| 79 | \( 1 + 146.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 134T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 190T + 9.40e3T^{2} \) |
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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
−10.93368520438387004591021462384, −10.34913422091247427965347447688, −9.576548967975523669263158284831, −8.379158771935378055289746023366, −7.17724177288283887180247738741, −6.04178419826801823165719356188, −5.36886853699501501833767542623, −4.66827396284034161390610368911, −2.34804608197085442419145609387, −1.30785955757393758233558516199,
1.30785955757393758233558516199, 2.34804608197085442419145609387, 4.66827396284034161390610368911, 5.36886853699501501833767542623, 6.04178419826801823165719356188, 7.17724177288283887180247738741, 8.379158771935378055289746023366, 9.576548967975523669263158284831, 10.34913422091247427965347447688, 10.93368520438387004591021462384
