| L(s) = 1 | + 0.933·2-s − 3-s − 1.12·4-s − 2.03·5-s − 0.933·6-s − 2.50·7-s − 2.92·8-s + 9-s − 1.89·10-s − 11-s + 1.12·12-s − 2.33·14-s + 2.03·15-s − 0.470·16-s − 3.69·17-s + 0.933·18-s + 0.177·19-s + 2.29·20-s + 2.50·21-s − 0.933·22-s − 4.83·23-s + 2.92·24-s − 0.872·25-s − 27-s + 2.82·28-s − 7.73·29-s + 1.89·30-s + ⋯ |
| L(s) = 1 | + 0.660·2-s − 0.577·3-s − 0.564·4-s − 0.908·5-s − 0.381·6-s − 0.946·7-s − 1.03·8-s + 0.333·9-s − 0.599·10-s − 0.301·11-s + 0.325·12-s − 0.624·14-s + 0.524·15-s − 0.117·16-s − 0.896·17-s + 0.220·18-s + 0.0408·19-s + 0.512·20-s + 0.546·21-s − 0.199·22-s − 1.00·23-s + 0.596·24-s − 0.174·25-s − 0.192·27-s + 0.534·28-s − 1.43·29-s + 0.346·30-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
| \(L(1)\) |
\(\approx\) |
\(0.002665351933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002665351933\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.933T + 2T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 - 0.177T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 + 8.83T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 + 0.511T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 0.823T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 + 9.81T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 + 7.01T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 8.91T + 97T^{2} \) |
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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
−8.205969288141821680573038276564, −7.16652439324765449850421289825, −6.74959093878513506848102979599, −5.68292338231820563754777778835, −5.40114690879973566466452639504, −4.33509288236681078957031182924, −3.80640793571783459459144970045, −3.26143476841031185278937653057, −1.94818977292572532227080255192, −0.02424552363627691087078375151,
0.02424552363627691087078375151, 1.94818977292572532227080255192, 3.26143476841031185278937653057, 3.80640793571783459459144970045, 4.33509288236681078957031182924, 5.40114690879973566466452639504, 5.68292338231820563754777778835, 6.74959093878513506848102979599, 7.16652439324765449850421289825, 8.205969288141821680573038276564
