| L(s) = 1 | − 2-s − 3-s + 4-s + 2.11·5-s + 6-s − 8-s + 9-s − 2.11·10-s + 3.35·11-s − 12-s − 4.23·13-s − 2.11·15-s + 16-s + 6.95·17-s − 18-s + 19-s + 2.11·20-s − 3.35·22-s − 7.18·23-s + 24-s − 0.513·25-s + 4.23·26-s − 27-s + 7.29·29-s + 2.11·30-s + 4.54·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.947·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.669·10-s + 1.01·11-s − 0.288·12-s − 1.17·13-s − 0.546·15-s + 0.250·16-s + 1.68·17-s − 0.235·18-s + 0.229·19-s + 0.473·20-s − 0.714·22-s − 1.49·23-s + 0.204·24-s − 0.102·25-s + 0.830·26-s − 0.192·27-s + 1.35·29-s + 0.386·30-s + 0.816·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473432494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473432494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.11T + 5T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 - 7.29T + 29T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 + 0.252T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 6.91T + 61T^{2} \) |
| 67 | \( 1 + 4.30T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 2.27T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 3.79T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
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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
−8.134987244618417208947409977758, −7.41195622412407003378974455715, −6.72107855248620335235367397020, −6.01635309145129081628754716982, −5.52016403888015839523414655578, −4.62776602917441108930849030494, −3.60968352552680732619742441402, −2.53943345156321214841740644385, −1.67534291264329115031398224968, −0.77426360056893783512585554240,
0.77426360056893783512585554240, 1.67534291264329115031398224968, 2.53943345156321214841740644385, 3.60968352552680732619742441402, 4.62776602917441108930849030494, 5.52016403888015839523414655578, 6.01635309145129081628754716982, 6.72107855248620335235367397020, 7.41195622412407003378974455715, 8.134987244618417208947409977758
