L(s) = 1 | + 2-s − 3-s + 4-s − 3.39·5-s − 6-s − 4.14·7-s + 8-s + 9-s − 3.39·10-s + 11-s − 12-s + 3.54·13-s − 4.14·14-s + 3.39·15-s + 16-s + 17-s + 18-s + 6.54·19-s − 3.39·20-s + 4.14·21-s + 22-s + 4.54·23-s − 24-s + 6.54·25-s + 3.54·26-s − 27-s − 4.14·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.51·5-s − 0.408·6-s − 1.56·7-s + 0.353·8-s + 0.333·9-s − 1.07·10-s + 0.301·11-s − 0.288·12-s + 0.982·13-s − 1.10·14-s + 0.877·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.50·19-s − 0.759·20-s + 0.904·21-s + 0.213·22-s + 0.947·23-s − 0.204·24-s + 1.30·25-s + 0.695·26-s − 0.192·27-s − 0.783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 6.54T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 + 9.14T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 - 0.497T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 + 6.69T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 67 | \( 1 - 9.14T + 67T^{2} \) |
| 71 | \( 1 + 9.14T + 71T^{2} \) |
| 73 | \( 1 - 4.64T + 73T^{2} \) |
| 79 | \( 1 - 3.54T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.82773311101280739098928041196, −7.04483914453183360312576505885, −6.75825759159854996255530425037, −5.75435964542530765566829335260, −5.15821540174972337854022879804, −3.99703069694419138957501151629, −3.56221318610584867510695219703, −3.02675117602580223500398283999, −1.21635534741111314794764307812, 0,
1.21635534741111314794764307812, 3.02675117602580223500398283999, 3.56221318610584867510695219703, 3.99703069694419138957501151629, 5.15821540174972337854022879804, 5.75435964542530765566829335260, 6.75825759159854996255530425037, 7.04483914453183360312576505885, 7.82773311101280739098928041196