| L(s) = 1 | + 2·3-s − 5-s + 9-s − 4·11-s − 2·15-s + 3·17-s + 2·19-s + 2·23-s − 4·25-s − 4·27-s + 5·29-s − 2·31-s − 8·33-s − 5·37-s − 3·41-s − 4·43-s − 45-s − 6·47-s − 7·49-s + 6·51-s + 13·53-s + 4·55-s + 4·57-s − 12·59-s − 7·61-s + 14·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.516·15-s + 0.727·17-s + 0.458·19-s + 0.417·23-s − 4/5·25-s − 0.769·27-s + 0.928·29-s − 0.359·31-s − 1.39·33-s − 0.821·37-s − 0.468·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s − 49-s + 0.840·51-s + 1.78·53-s + 0.539·55-s + 0.529·57-s − 1.56·59-s − 0.896·61-s + 1.71·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.80089797869998822407567424466, −7.50011611270933841721970247035, −6.50872443620216129464181972797, −5.49323949027454266462783399032, −4.91630660148033549461132254440, −3.83587750033311883560048490711, −3.19088788707795168129396983529, −2.59194700591872604992999629813, −1.54311899631223698230214774056, 0,
1.54311899631223698230214774056, 2.59194700591872604992999629813, 3.19088788707795168129396983529, 3.83587750033311883560048490711, 4.91630660148033549461132254440, 5.49323949027454266462783399032, 6.50872443620216129464181972797, 7.50011611270933841721970247035, 7.80089797869998822407567424466
