L(s) = 1 | − 0.349·2-s − 0.991·3-s − 1.87·4-s + 5-s + 0.346·6-s − 3.74·7-s + 1.35·8-s − 2.01·9-s − 0.349·10-s + 0.640·11-s + 1.86·12-s + 0.642·13-s + 1.30·14-s − 0.991·15-s + 3.28·16-s + 3.58·17-s + 0.704·18-s + 6.99·19-s − 1.87·20-s + 3.71·21-s − 0.223·22-s + 1.75·23-s − 1.34·24-s + 25-s − 0.224·26-s + 4.97·27-s + 7.03·28-s + ⋯ |
L(s) = 1 | − 0.246·2-s − 0.572·3-s − 0.939·4-s + 0.447·5-s + 0.141·6-s − 1.41·7-s + 0.478·8-s − 0.672·9-s − 0.110·10-s + 0.193·11-s + 0.537·12-s + 0.178·13-s + 0.349·14-s − 0.255·15-s + 0.820·16-s + 0.869·17-s + 0.166·18-s + 1.60·19-s − 0.419·20-s + 0.810·21-s − 0.0476·22-s + 0.365·23-s − 0.273·24-s + 0.200·25-s − 0.0439·26-s + 0.957·27-s + 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + 0.349T + 2T^{2} \) |
| 3 | \( 1 + 0.991T + 3T^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 0.640T + 11T^{2} \) |
| 13 | \( 1 - 0.642T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 6.99T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 + 6.69T + 31T^{2} \) |
| 37 | \( 1 + 6.14T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 - 8.12T + 53T^{2} \) |
| 59 | \( 1 + 7.01T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 7.38T + 71T^{2} \) |
| 73 | \( 1 - 9.56T + 73T^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 - 9.93T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 + 7.55T + 97T^{2} \) |
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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
−8.993211599931761621278071204298, −8.127042264932802435559429100439, −7.10202102795833867801812449293, −6.30243305294035450650265142277, −5.48935783422594925604913305985, −5.01341195492336734025443434830, −3.55608054372903449568044769401, −3.07629895488873573452980703385, −1.21170269318379368662503243488, 0,
1.21170269318379368662503243488, 3.07629895488873573452980703385, 3.55608054372903449568044769401, 5.01341195492336734025443434830, 5.48935783422594925604913305985, 6.30243305294035450650265142277, 7.10202102795833867801812449293, 8.127042264932802435559429100439, 8.993211599931761621278071204298