L(s) = 1 | + 3·3-s + 4·4-s − 8.66·7-s + 9·9-s + 12·12-s + 13.8·13-s + 16·16-s + 36.3·19-s − 25.9·21-s + 25·25-s + 27·27-s − 34.6·28-s − 59·31-s + 36·36-s − 47·37-s + 41.5·39-s − 83.1·43-s + 48·48-s + 26.0·49-s + 55.4·52-s + 109.·57-s − 15.5·61-s − 77.9·63-s + 64·64-s + 13·67-s + 29.4·73-s + 75·75-s + ⋯ |
L(s) = 1 | + 3-s + 4-s − 1.23·7-s + 9-s + 12-s + 1.06·13-s + 16-s + 1.91·19-s − 1.23·21-s + 25-s + 27-s − 1.23·28-s − 1.90·31-s + 36-s − 1.27·37-s + 1.06·39-s − 1.93·43-s + 48-s + 0.530·49-s + 1.06·52-s + 1.91·57-s − 0.255·61-s − 1.23·63-s + 64-s + 0.194·67-s + 0.403·73-s + 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.829229039\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.829229039\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 4T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 + 8.66T + 49T^{2} \) |
| 13 | \( 1 - 13.8T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 36.3T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 59T + 961T^{2} \) |
| 37 | \( 1 + 47T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 83.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 13T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 29.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 157.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 169T + 9.40e3T^{2} \) |
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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
−11.13603096951249779300282852858, −10.17880938598833176776962579337, −9.402394231522816330464951057032, −8.447880683512287180381504150278, −7.27385793035487535678109372188, −6.72423690637824941369700314632, −5.49256043166584794127852791097, −3.52969422957625937247857782719, −3.07944644262946848786602618899, −1.50839786611026467460021403066,
1.50839786611026467460021403066, 3.07944644262946848786602618899, 3.52969422957625937247857782719, 5.49256043166584794127852791097, 6.72423690637824941369700314632, 7.27385793035487535678109372188, 8.447880683512287180381504150278, 9.402394231522816330464951057032, 10.17880938598833176776962579337, 11.13603096951249779300282852858