L(s) = 1 | − 1.93·3-s + 2.25·5-s − 7-s + 0.745·9-s − 11-s + 13-s − 4.36·15-s + 0.616·17-s + 2.42·19-s + 1.93·21-s − 7.69·23-s + 0.0809·25-s + 4.36·27-s + 8.37·29-s − 3.31·31-s + 1.93·33-s − 2.25·35-s + 4.68·37-s − 1.93·39-s − 12.2·41-s + 2.23·43-s + 1.68·45-s − 4.85·47-s + 49-s − 1.19·51-s + 6.99·53-s − 2.25·55-s + ⋯ |
L(s) = 1 | − 1.11·3-s + 1.00·5-s − 0.377·7-s + 0.248·9-s − 0.301·11-s + 0.277·13-s − 1.12·15-s + 0.149·17-s + 0.556·19-s + 0.422·21-s − 1.60·23-s + 0.0161·25-s + 0.839·27-s + 1.55·29-s − 0.596·31-s + 0.336·33-s − 0.381·35-s + 0.769·37-s − 0.309·39-s − 1.90·41-s + 0.341·43-s + 0.250·45-s − 0.708·47-s + 0.142·49-s − 0.167·51-s + 0.960·53-s − 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 - 2.25T + 5T^{2} \) |
| 17 | \( 1 - 0.616T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 + 7.69T + 23T^{2} \) |
| 29 | \( 1 - 8.37T + 29T^{2} \) |
| 31 | \( 1 + 3.31T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 - 6.99T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 2.74T + 61T^{2} \) |
| 67 | \( 1 + 5.42T + 67T^{2} \) |
| 71 | \( 1 + 2.38T + 71T^{2} \) |
| 73 | \( 1 + 4.98T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 3.95T + 83T^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.29932741492675772219353689073, −6.50955220488829500330168221975, −6.06734875968744111386927469730, −5.53078847487313590767644337373, −4.95101429408696187818060667979, −4.01280813357970430282356885122, −3.02755822789317687495521529885, −2.14781565556594649594445845737, −1.14821143967501402151942184797, 0,
1.14821143967501402151942184797, 2.14781565556594649594445845737, 3.02755822789317687495521529885, 4.01280813357970430282356885122, 4.95101429408696187818060667979, 5.53078847487313590767644337373, 6.06734875968744111386927469730, 6.50955220488829500330168221975, 7.29932741492675772219353689073