| L(s) = 1 | − 2·3-s + 9-s + 4·11-s + 4·13-s + 2·17-s − 6·19-s − 8·23-s − 5·25-s + 4·27-s + 2·29-s − 4·31-s − 8·33-s + 10·37-s − 8·39-s + 10·41-s − 4·43-s + 4·47-s − 4·51-s − 2·53-s + 12·57-s + 10·59-s + 8·61-s + 8·67-s + 16·69-s + 6·73-s + 10·75-s + 16·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 0.485·17-s − 1.37·19-s − 1.66·23-s − 25-s + 0.769·27-s + 0.371·29-s − 0.718·31-s − 1.39·33-s + 1.64·37-s − 1.28·39-s + 1.56·41-s − 0.609·43-s + 0.583·47-s − 0.560·51-s − 0.274·53-s + 1.58·57-s + 1.30·59-s + 1.02·61-s + 0.977·67-s + 1.92·69-s + 0.702·73-s + 1.15·75-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098243670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098243670\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.549075488975808296460771323258, −8.599368519419079262478379431683, −7.890116261146935042213485652407, −6.64750878269403183094013571175, −6.13738869871460904432513586375, −5.61922713728706138617525130188, −4.29163180327559437867512990822, −3.78335663933465066778564635104, −2.09895130048690670822747691948, −0.793259487937152966248793400164,
0.793259487937152966248793400164, 2.09895130048690670822747691948, 3.78335663933465066778564635104, 4.29163180327559437867512990822, 5.61922713728706138617525130188, 6.13738869871460904432513586375, 6.64750878269403183094013571175, 7.890116261146935042213485652407, 8.599368519419079262478379431683, 9.549075488975808296460771323258
