| L(s) = 1 | − 3-s + 9-s − 5·11-s − 4·13-s − 2·17-s − 4·19-s + 9·23-s − 27-s − 5·29-s − 2·31-s + 5·33-s − 11·37-s + 4·39-s − 8·41-s − 11·43-s − 12·47-s + 2·51-s − 10·53-s + 4·57-s − 12·59-s + 12·61-s + 3·67-s − 9·69-s + 3·71-s + 6·73-s + 7·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.50·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s + 1.87·23-s − 0.192·27-s − 0.928·29-s − 0.359·31-s + 0.870·33-s − 1.80·37-s + 0.640·39-s − 1.24·41-s − 1.67·43-s − 1.75·47-s + 0.280·51-s − 1.37·53-s + 0.529·57-s − 1.56·59-s + 1.53·61-s + 0.366·67-s − 1.08·69-s + 0.356·71-s + 0.702·73-s + 0.787·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.55240985584039, −15.14946437080697, −14.92505514203506, −14.06233903511351, −13.38606509259650, −12.94710984178101, −12.63614864328246, −12.02232677653742, −11.22817934163483, −10.94945954978478, −10.41133738165588, −9.816987627277708, −9.341882692479352, −8.440870249531853, −8.157535059933474, −7.253499893440336, −6.919840327428827, −6.369598101208724, −5.341650954931674, −5.036989979738781, −4.754054522331546, −3.598913580771330, −3.024520078249347, −2.191382318711992, −1.557053846239484, 0, 0,
1.557053846239484, 2.191382318711992, 3.024520078249347, 3.598913580771330, 4.754054522331546, 5.036989979738781, 5.341650954931674, 6.369598101208724, 6.919840327428827, 7.253499893440336, 8.157535059933474, 8.440870249531853, 9.341882692479352, 9.816987627277708, 10.41133738165588, 10.94945954978478, 11.22817934163483, 12.02232677653742, 12.63614864328246, 12.94710984178101, 13.38606509259650, 14.06233903511351, 14.92505514203506, 15.14946437080697, 15.55240985584039
