| L(s) = 1 | + 0.847·2-s − 3-s − 1.28·4-s + 2.90·5-s − 0.847·6-s + 4.44·7-s − 2.78·8-s + 9-s + 2.46·10-s + 11-s + 1.28·12-s + 3.76·14-s − 2.90·15-s + 0.203·16-s − 2.03·17-s + 0.847·18-s + 2.98·19-s − 3.72·20-s − 4.44·21-s + 0.847·22-s − 6.63·23-s + 2.78·24-s + 3.46·25-s − 27-s − 5.69·28-s + 4.50·29-s − 2.46·30-s + ⋯ |
| L(s) = 1 | + 0.599·2-s − 0.577·3-s − 0.640·4-s + 1.30·5-s − 0.346·6-s + 1.67·7-s − 0.983·8-s + 0.333·9-s + 0.780·10-s + 0.301·11-s + 0.369·12-s + 1.00·14-s − 0.751·15-s + 0.0509·16-s − 0.493·17-s + 0.199·18-s + 0.684·19-s − 0.833·20-s − 0.969·21-s + 0.180·22-s − 1.38·23-s + 0.567·24-s + 0.693·25-s − 0.192·27-s − 1.07·28-s + 0.835·29-s − 0.450·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.983332625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.983332625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.847T + 2T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 - 2.98T + 19T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 6.66T + 37T^{2} \) |
| 41 | \( 1 + 6.06T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 - 2.45T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 + 3.29T + 67T^{2} \) |
| 71 | \( 1 + 9.62T + 71T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 - 9.34T + 79T^{2} \) |
| 83 | \( 1 + 8.98T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 - 1.02T + 97T^{2} \) |
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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
−8.242091011626567305310186471727, −7.39551700194010206352485020282, −6.35613934911350242640243295097, −5.76654811506372379251429657530, −5.33210898208764476308840077860, −4.51189916887813730503160464460, −4.14967056297365204683233716189, −2.72196709062270928604284440960, −1.83569944059681080578080656030, −0.931382936965114407020189465529,
0.931382936965114407020189465529, 1.83569944059681080578080656030, 2.72196709062270928604284440960, 4.14967056297365204683233716189, 4.51189916887813730503160464460, 5.33210898208764476308840077860, 5.76654811506372379251429657530, 6.35613934911350242640243295097, 7.39551700194010206352485020282, 8.242091011626567305310186471727
