| L(s) = 1 | + 1.30·2-s − 3·3-s − 6.30·4-s − 8.10·5-s − 3.90·6-s + 0.543·7-s − 18.6·8-s + 9·9-s − 10.5·10-s + 18.9·12-s − 69.6·13-s + 0.708·14-s + 24.3·15-s + 26.1·16-s − 42.5·17-s + 11.7·18-s + 75.4·19-s + 51.0·20-s − 1.63·21-s + 37.9·23-s + 55.9·24-s − 59.3·25-s − 90.7·26-s − 27·27-s − 3.42·28-s + 120.·29-s + 31.6·30-s + ⋯ |
| L(s) = 1 | + 0.460·2-s − 0.577·3-s − 0.787·4-s − 0.724·5-s − 0.266·6-s + 0.0293·7-s − 0.823·8-s + 0.333·9-s − 0.333·10-s + 0.454·12-s − 1.48·13-s + 0.0135·14-s + 0.418·15-s + 0.408·16-s − 0.607·17-s + 0.153·18-s + 0.911·19-s + 0.570·20-s − 0.0169·21-s + 0.344·23-s + 0.475·24-s − 0.474·25-s − 0.684·26-s − 0.192·27-s − 0.0231·28-s + 0.769·29-s + 0.192·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9224952968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9224952968\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.30T + 8T^{2} \) |
| 5 | \( 1 + 8.10T + 125T^{2} \) |
| 7 | \( 1 - 0.543T + 343T^{2} \) |
| 13 | \( 1 + 69.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 37.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 277.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 14.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 284.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 312.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 337.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 169.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 801.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 386.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 981.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 535.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 456.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 619.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 198.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 929.T + 9.12e5T^{2} \) |
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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
−11.22304643575786409354479831196, −10.03055327627824363478749054234, −9.291825301385411864976315812084, −8.096067152551807838519433738904, −7.20766342809740409554402200431, −5.95894874818527679497581596349, −4.85890829297525469661207323347, −4.27056644410380923377571668619, −2.85415315862051719463443411640, −0.60534007524731996805084781407,
0.60534007524731996805084781407, 2.85415315862051719463443411640, 4.27056644410380923377571668619, 4.85890829297525469661207323347, 5.95894874818527679497581596349, 7.20766342809740409554402200431, 8.096067152551807838519433738904, 9.291825301385411864976315812084, 10.03055327627824363478749054234, 11.22304643575786409354479831196
