L(s) = 1 | − 2-s − 1.77·3-s + 4-s − 5-s + 1.77·6-s − 3.83·7-s − 8-s + 0.135·9-s + 10-s + 4.58·11-s − 1.77·12-s + 3.83·14-s + 1.77·15-s + 16-s + 3.86·17-s − 0.135·18-s − 5.15·19-s − 20-s + 6.79·21-s − 4.58·22-s + 4.98·23-s + 1.77·24-s + 25-s + 5.07·27-s − 3.83·28-s + 0.976·29-s − 1.77·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.02·3-s + 0.5·4-s − 0.447·5-s + 0.722·6-s − 1.45·7-s − 0.353·8-s + 0.0450·9-s + 0.316·10-s + 1.38·11-s − 0.511·12-s + 1.02·14-s + 0.457·15-s + 0.250·16-s + 0.938·17-s − 0.0318·18-s − 1.18·19-s − 0.223·20-s + 1.48·21-s − 0.976·22-s + 1.03·23-s + 0.361·24-s + 0.200·25-s + 0.976·27-s − 0.725·28-s + 0.181·29-s − 0.323·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 1.77T + 3T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 17 | \( 1 - 3.86T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 - 0.976T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 - 2.57T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 + 0.238T + 53T^{2} \) |
| 59 | \( 1 + 1.77T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 4.50T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 2.09T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−8.913448199924059801652350257713, −8.388265621849923123844552698875, −7.06080201998919975164963021757, −6.55762624994321393796598936779, −6.08291869174060322353361017318, −4.93488821500285801746883984427, −3.74589066663298996283876519899, −2.91144588868274223035256519312, −1.15847136066301040393879022357, 0,
1.15847136066301040393879022357, 2.91144588868274223035256519312, 3.74589066663298996283876519899, 4.93488821500285801746883984427, 6.08291869174060322353361017318, 6.55762624994321393796598936779, 7.06080201998919975164963021757, 8.388265621849923123844552698875, 8.913448199924059801652350257713