L(s) = 1 | − 2-s + 3.27·3-s + 4-s + 0.672·5-s − 3.27·6-s − 2.46·7-s − 8-s + 7.74·9-s − 0.672·10-s − 2.72·11-s + 3.27·12-s − 4.85·13-s + 2.46·14-s + 2.20·15-s + 16-s − 5.69·17-s − 7.74·18-s + 4.50·19-s + 0.672·20-s − 8.09·21-s + 2.72·22-s − 23-s − 3.27·24-s − 4.54·25-s + 4.85·26-s + 15.5·27-s − 2.46·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.89·3-s + 0.5·4-s + 0.300·5-s − 1.33·6-s − 0.933·7-s − 0.353·8-s + 2.58·9-s − 0.212·10-s − 0.820·11-s + 0.946·12-s − 1.34·13-s + 0.659·14-s + 0.569·15-s + 0.250·16-s − 1.38·17-s − 1.82·18-s + 1.03·19-s + 0.150·20-s − 1.76·21-s + 0.579·22-s − 0.208·23-s − 0.668·24-s − 0.909·25-s + 0.952·26-s + 2.99·27-s − 0.466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 3.27T + 3T^{2} \) |
| 5 | \( 1 - 0.672T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 29 | \( 1 + 0.111T + 29T^{2} \) |
| 31 | \( 1 - 0.359T + 31T^{2} \) |
| 37 | \( 1 - 3.00T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 - 4.79T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 + 0.654T + 61T^{2} \) |
| 67 | \( 1 + 0.983T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 8.31T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 3.35T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 9.65T + 97T^{2} \) |
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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
−7.947439369055605359251871590754, −7.09491058517119825774728550794, −6.87648979877738163739533862848, −5.65881366875758050178889687322, −4.62812917836719298244163431833, −3.75174280427556281061563316708, −2.79082784679866247875160570608, −2.54044496831589407334097789043, −1.64228023370119954292412843957, 0,
1.64228023370119954292412843957, 2.54044496831589407334097789043, 2.79082784679866247875160570608, 3.75174280427556281061563316708, 4.62812917836719298244163431833, 5.65881366875758050178889687322, 6.87648979877738163739533862848, 7.09491058517119825774728550794, 7.947439369055605359251871590754