| L(s) = 1 | − 2-s + 3-s + 4-s + 3.15·5-s − 6-s − 7-s − 8-s + 9-s − 3.15·10-s + 1.09·11-s + 12-s + 4.65·13-s + 14-s + 3.15·15-s + 16-s + 2.27·17-s − 18-s + 1.84·19-s + 3.15·20-s − 21-s − 1.09·22-s − 6.79·23-s − 24-s + 4.96·25-s − 4.65·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.41·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.998·10-s + 0.330·11-s + 0.288·12-s + 1.29·13-s + 0.267·14-s + 0.814·15-s + 0.250·16-s + 0.552·17-s − 0.235·18-s + 0.423·19-s + 0.705·20-s − 0.218·21-s − 0.233·22-s − 1.41·23-s − 0.204·24-s + 0.992·25-s − 0.913·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.842221851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.842221851\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 5 | \( 1 - 3.15T + 5T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 + 6.79T + 23T^{2} \) |
| 29 | \( 1 - 5.10T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 + 4.73T + 37T^{2} \) |
| 41 | \( 1 + 4.15T + 41T^{2} \) |
| 43 | \( 1 - 5.56T + 43T^{2} \) |
| 47 | \( 1 + 4.09T + 47T^{2} \) |
| 53 | \( 1 - 4.60T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 6.78T + 61T^{2} \) |
| 67 | \( 1 - 4.14T + 67T^{2} \) |
| 71 | \( 1 - 0.443T + 71T^{2} \) |
| 73 | \( 1 - 0.783T + 73T^{2} \) |
| 79 | \( 1 + 5.90T + 79T^{2} \) |
| 83 | \( 1 - 7.66T + 83T^{2} \) |
| 89 | \( 1 + 1.36T + 89T^{2} \) |
| 97 | \( 1 + 0.948T + 97T^{2} \) |
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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
−8.170970160804358156168462355918, −7.10625475028641556025859731584, −6.41672295650710503180733222023, −6.01411777294767640585549901363, −5.26903674124542223883151809947, −4.09762187062831198018662853723, −3.30107565892366711062435864269, −2.50449509522315859867325612978, −1.69679429055679659078169925974, −0.970099098795724651062694500951,
0.970099098795724651062694500951, 1.69679429055679659078169925974, 2.50449509522315859867325612978, 3.30107565892366711062435864269, 4.09762187062831198018662853723, 5.26903674124542223883151809947, 6.01411777294767640585549901363, 6.41672295650710503180733222023, 7.10625475028641556025859731584, 8.170970160804358156168462355918
