| L(s) = 1 | + 3·3-s + 7-s + 9·9-s − 42·11-s − 67·13-s + 54·17-s + 115·19-s + 3·21-s + 162·23-s + 27·27-s − 210·29-s + 193·31-s − 126·33-s − 286·37-s − 201·39-s + 12·41-s − 263·43-s − 414·47-s − 342·49-s + 162·51-s − 192·53-s + 345·57-s − 690·59-s − 733·61-s + 9·63-s − 299·67-s + 486·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0539·7-s + 1/3·9-s − 1.15·11-s − 1.42·13-s + 0.770·17-s + 1.38·19-s + 0.0311·21-s + 1.46·23-s + 0.192·27-s − 1.34·29-s + 1.11·31-s − 0.664·33-s − 1.27·37-s − 0.825·39-s + 0.0457·41-s − 0.932·43-s − 1.28·47-s − 0.997·49-s + 0.444·51-s − 0.497·53-s + 0.801·57-s − 1.52·59-s − 1.53·61-s + 0.0179·63-s − 0.545·67-s + 0.847·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 42 T + p^{3} T^{2} \) |
| 13 | \( 1 + 67 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 115 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 - 193 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 - 12 T + p^{3} T^{2} \) |
| 43 | \( 1 + 263 T + p^{3} T^{2} \) |
| 47 | \( 1 + 414 T + p^{3} T^{2} \) |
| 53 | \( 1 + 192 T + p^{3} T^{2} \) |
| 59 | \( 1 + 690 T + p^{3} T^{2} \) |
| 61 | \( 1 + 733 T + p^{3} T^{2} \) |
| 67 | \( 1 + 299 T + p^{3} T^{2} \) |
| 71 | \( 1 - 228 T + p^{3} T^{2} \) |
| 73 | \( 1 - 938 T + p^{3} T^{2} \) |
| 79 | \( 1 - 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 462 T + p^{3} T^{2} \) |
| 89 | \( 1 + 240 T + p^{3} T^{2} \) |
| 97 | \( 1 + 511 T + p^{3} T^{2} \) |
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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
−9.100548403124672527552890993560, −7.82764562945355252089901481936, −7.65968274306420625499802463810, −6.64223106338031832985161242832, −5.17980363032097643681166099681, −4.96433254377017234594067223240, −3.34396641954505354427343965194, −2.78094320410323282978862001175, −1.51077511291527613084061952425, 0,
1.51077511291527613084061952425, 2.78094320410323282978862001175, 3.34396641954505354427343965194, 4.96433254377017234594067223240, 5.17980363032097643681166099681, 6.64223106338031832985161242832, 7.65968274306420625499802463810, 7.82764562945355252089901481936, 9.100548403124672527552890993560
