| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 7-s + 8-s + 9-s − 3·11-s + 2·12-s + 13-s − 14-s + 16-s + 18-s + 19-s − 2·21-s − 3·22-s + 2·24-s + 26-s − 4·27-s − 28-s − 3·29-s + 2·31-s + 32-s − 6·33-s + 36-s + 2·37-s + 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.229·19-s − 0.436·21-s − 0.639·22-s + 0.408·24-s + 0.196·26-s − 0.769·27-s − 0.188·28-s − 0.557·29-s + 0.359·31-s + 0.176·32-s − 1.04·33-s + 1/6·36-s + 0.328·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.50047387942535, −14.97003401782702, −14.37270915174990, −14.01079345999779, −13.48080675117712, −12.94732194839318, −12.69568459254834, −11.88883442173006, −11.23238585334106, −10.81581688968282, −10.01552285727573, −9.586518015776864, −8.960717753546329, −8.340012548495449, −7.714066784122482, −7.484086324316888, −6.476702015876215, −6.075595718353459, −5.260182025846178, −4.727566551943978, −3.865363412788118, −3.357514461525315, −2.781876990070150, −2.240114394715300, −1.364715054446121, 0,
1.364715054446121, 2.240114394715300, 2.781876990070150, 3.357514461525315, 3.865363412788118, 4.727566551943978, 5.260182025846178, 6.075595718353459, 6.476702015876215, 7.484086324316888, 7.714066784122482, 8.340012548495449, 8.960717753546329, 9.586518015776864, 10.01552285727573, 10.81581688968282, 11.23238585334106, 11.88883442173006, 12.69568459254834, 12.94732194839318, 13.48080675117712, 14.01079345999779, 14.37270915174990, 14.97003401782702, 15.50047387942535
