| L(s) = 1 | − 0.881·2-s + 3-s − 1.22·4-s − 2.92·5-s − 0.881·6-s + 2.84·8-s + 9-s + 2.58·10-s + 3.24·11-s − 1.22·12-s + 1.62·13-s − 2.92·15-s − 0.0586·16-s + 4.45·17-s − 0.881·18-s − 1.70·19-s + 3.57·20-s − 2.86·22-s − 23-s + 2.84·24-s + 3.56·25-s − 1.43·26-s + 27-s − 8.39·29-s + 2.58·30-s + 3.59·31-s − 5.63·32-s + ⋯ |
| L(s) = 1 | − 0.623·2-s + 0.577·3-s − 0.611·4-s − 1.30·5-s − 0.359·6-s + 1.00·8-s + 0.333·9-s + 0.815·10-s + 0.979·11-s − 0.353·12-s + 0.451·13-s − 0.755·15-s − 0.0146·16-s + 1.08·17-s − 0.207·18-s − 0.390·19-s + 0.800·20-s − 0.610·22-s − 0.208·23-s + 0.579·24-s + 0.713·25-s − 0.281·26-s + 0.192·27-s − 1.55·29-s + 0.471·30-s + 0.646·31-s − 0.995·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.060217759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060217759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.881T + 2T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 29 | \( 1 + 8.39T + 29T^{2} \) |
| 31 | \( 1 - 3.59T + 31T^{2} \) |
| 37 | \( 1 + 1.06T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 5.36T + 47T^{2} \) |
| 53 | \( 1 + 0.262T + 53T^{2} \) |
| 59 | \( 1 - 4.00T + 59T^{2} \) |
| 61 | \( 1 - 3.00T + 61T^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 - 0.422T + 73T^{2} \) |
| 79 | \( 1 - 0.231T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 0.971T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
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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
−8.463794838767232284212621997927, −8.059464076319142733858191773810, −7.47174630949582680234173640748, −6.66686624645421914093479710023, −5.52269620099981666679899773088, −4.47206263345057947292243381418, −3.85252852472737572492729287010, −3.34792990989792770201044661282, −1.75362546081299381716595901714, −0.68583916633256404655013082876,
0.68583916633256404655013082876, 1.75362546081299381716595901714, 3.34792990989792770201044661282, 3.85252852472737572492729287010, 4.47206263345057947292243381418, 5.52269620099981666679899773088, 6.66686624645421914093479710023, 7.47174630949582680234173640748, 8.059464076319142733858191773810, 8.463794838767232284212621997927
