L(s) = 1 | − 2-s + 4-s − 1.20·5-s + 4.22·7-s − 8-s + 1.20·10-s − 1.67·11-s − 3.39·13-s − 4.22·14-s + 16-s − 6.30·17-s + 0.549·19-s − 1.20·20-s + 1.67·22-s + 1.62·23-s − 3.54·25-s + 3.39·26-s + 4.22·28-s + 5.50·29-s + 5.74·31-s − 32-s + 6.30·34-s − 5.08·35-s + 6.81·37-s − 0.549·38-s + 1.20·40-s + 1.19·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.538·5-s + 1.59·7-s − 0.353·8-s + 0.381·10-s − 0.506·11-s − 0.941·13-s − 1.12·14-s + 0.250·16-s − 1.52·17-s + 0.126·19-s − 0.269·20-s + 0.357·22-s + 0.338·23-s − 0.709·25-s + 0.665·26-s + 0.797·28-s + 1.02·29-s + 1.03·31-s − 0.176·32-s + 1.08·34-s − 0.859·35-s + 1.12·37-s − 0.0892·38-s + 0.190·40-s + 0.185·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 + 1.20T + 5T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 + 6.30T + 17T^{2} \) |
| 19 | \( 1 - 0.549T + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 - 1.19T + 41T^{2} \) |
| 43 | \( 1 + 3.93T + 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 + 1.95T + 59T^{2} \) |
| 61 | \( 1 - 0.935T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 - 2.12T + 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 1.69T + 83T^{2} \) |
| 89 | \( 1 - 1.92T + 89T^{2} \) |
| 97 | \( 1 + 0.826T + 97T^{2} \) |
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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
−7.78319008079162593227904143133, −6.97839028816774570172633488208, −6.31355383123612819485736003536, −5.20612190751117667148760674547, −4.71523278335196147627227900710, −4.08271633749715402181836347461, −2.74239392654674971223691472383, −2.20224546773035508332562250269, −1.18619473954663238574352339496, 0,
1.18619473954663238574352339496, 2.20224546773035508332562250269, 2.74239392654674971223691472383, 4.08271633749715402181836347461, 4.71523278335196147627227900710, 5.20612190751117667148760674547, 6.31355383123612819485736003536, 6.97839028816774570172633488208, 7.78319008079162593227904143133