L(s) = 1 | + 2-s − 0.142·3-s + 4-s − 5-s − 0.142·6-s − 2.21·7-s + 8-s − 2.97·9-s − 10-s − 11-s − 0.142·12-s − 2.80·13-s − 2.21·14-s + 0.142·15-s + 16-s + 6.32·17-s − 2.97·18-s + 7.66·19-s − 20-s + 0.314·21-s − 22-s + 3.01·23-s − 0.142·24-s + 25-s − 2.80·26-s + 0.850·27-s − 2.21·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.0821·3-s + 0.5·4-s − 0.447·5-s − 0.0580·6-s − 0.836·7-s + 0.353·8-s − 0.993·9-s − 0.316·10-s − 0.301·11-s − 0.0410·12-s − 0.777·13-s − 0.591·14-s + 0.0367·15-s + 0.250·16-s + 1.53·17-s − 0.702·18-s + 1.75·19-s − 0.223·20-s + 0.0686·21-s − 0.213·22-s + 0.628·23-s − 0.0290·24-s + 0.200·25-s − 0.550·26-s + 0.163·27-s − 0.418·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
good | 3 | \( 1 + 0.142T + 3T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 - 3.01T + 23T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 31 | \( 1 - 8.72T + 31T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 + 9.30T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 - 0.942T + 47T^{2} \) |
| 53 | \( 1 + 6.60T + 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 7.32T + 67T^{2} \) |
| 71 | \( 1 + 4.92T + 71T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.57029244687384003968843341973, −6.70731942376687457933051706601, −6.01117627120820664505973818385, −5.19121800903351671511998219767, −4.99785315440912262628327715710, −3.62046702911887395600392662715, −3.20860129631183308492497404719, −2.67211193823051912765077949137, −1.25018189660373457754011457117, 0,
1.25018189660373457754011457117, 2.67211193823051912765077949137, 3.20860129631183308492497404719, 3.62046702911887395600392662715, 4.99785315440912262628327715710, 5.19121800903351671511998219767, 6.01117627120820664505973818385, 6.70731942376687457933051706601, 7.57029244687384003968843341973