L(s) = 1 | + 1.73·3-s − 4i·5-s − 6.92i·7-s + 2.99·9-s + 6.92·11-s − 6.92i·15-s − 18·17-s − 20.7·19-s − 11.9i·21-s − 41.5i·23-s + 9·25-s + 5.19·27-s − 4i·29-s − 48.4i·31-s + 11.9·33-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.800i·5-s − 0.989i·7-s + 0.333·9-s + 0.629·11-s − 0.461i·15-s − 1.05·17-s − 1.09·19-s − 0.571i·21-s − 1.80i·23-s + 0.359·25-s + 0.192·27-s − 0.137i·29-s − 1.56i·31-s + 0.363·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31814 - 1.31814i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31814 - 1.31814i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
good | 5 | \( 1 + 4iT - 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 6.92T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 18T + 289T^{2} \) |
| 19 | \( 1 + 20.7T + 361T^{2} \) |
| 23 | \( 1 + 41.5iT - 529T^{2} \) |
| 29 | \( 1 + 4iT - 841T^{2} \) |
| 31 | \( 1 + 48.4iT - 961T^{2} \) |
| 37 | \( 1 - 72iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 44iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 72iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 20.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 41.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 - 62.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 131.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 126T + 7.92e3T^{2} \) |
| 97 | \( 1 - 110T + 9.40e3T^{2} \) |
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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
−10.77781914922967706044788496568, −9.962654625382763741805347017886, −8.839026975294837554068814851676, −8.375378454168037238430867973446, −7.12833485260679962922957985679, −6.27392022834307555599641346769, −4.55049271045132023628077433739, −4.09998419567446782834805318384, −2.36685497314927407665575740552, −0.78042646156902315649360551355,
1.95232507248799730840320909591, 3.02454273033745894996260135960, 4.22003201048297053337997931721, 5.68459333959429084492062827074, 6.68087479353083523481473786644, 7.56553605493569526247158035262, 8.907462226000912884087882426527, 9.168308817194843852867729519374, 10.55151378329355431683288711622, 11.22979279468012920180613391025