| L(s) = 1 | + 18·7-s + 16·11-s + 6·13-s − 6·17-s − 124·19-s + 42·23-s − 142·29-s − 188·31-s − 202·37-s − 54·41-s − 66·43-s + 38·47-s − 19·49-s + 738·53-s − 564·59-s − 262·61-s + 554·67-s − 140·71-s − 882·73-s + 288·77-s − 1.16e3·79-s + 642·83-s + 854·89-s + 108·91-s + 478·97-s + 1.79e3·101-s − 642·103-s + ⋯ |
| L(s) = 1 | + 0.971·7-s + 0.438·11-s + 0.128·13-s − 0.0856·17-s − 1.49·19-s + 0.380·23-s − 0.909·29-s − 1.08·31-s − 0.897·37-s − 0.205·41-s − 0.234·43-s + 0.117·47-s − 0.0553·49-s + 1.91·53-s − 1.24·59-s − 0.549·61-s + 1.01·67-s − 0.234·71-s − 1.41·73-s + 0.426·77-s − 1.65·79-s + 0.849·83-s + 1.01·89-s + 0.124·91-s + 0.500·97-s + 1.76·101-s − 0.614·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 142 T + p^{3} T^{2} \) |
| 31 | \( 1 + 188 T + p^{3} T^{2} \) |
| 37 | \( 1 + 202 T + p^{3} T^{2} \) |
| 41 | \( 1 + 54 T + p^{3} T^{2} \) |
| 43 | \( 1 + 66 T + p^{3} T^{2} \) |
| 47 | \( 1 - 38 T + p^{3} T^{2} \) |
| 53 | \( 1 - 738 T + p^{3} T^{2} \) |
| 59 | \( 1 + 564 T + p^{3} T^{2} \) |
| 61 | \( 1 + 262 T + p^{3} T^{2} \) |
| 67 | \( 1 - 554 T + p^{3} T^{2} \) |
| 71 | \( 1 + 140 T + p^{3} T^{2} \) |
| 73 | \( 1 + 882 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 642 T + p^{3} T^{2} \) |
| 89 | \( 1 - 854 T + p^{3} T^{2} \) |
| 97 | \( 1 - 478 T + p^{3} T^{2} \) |
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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
−8.677032330280436954302927106022, −7.73151981954266855813602480501, −7.00437159362150361684871153334, −6.09501456817915574557965042986, −5.22087021827547910939606776794, −4.37983659672026445232459538768, −3.56884531776830265114822156562, −2.21921290175955667938471934862, −1.43657579461213153138551171854, 0,
1.43657579461213153138551171854, 2.21921290175955667938471934862, 3.56884531776830265114822156562, 4.37983659672026445232459538768, 5.22087021827547910939606776794, 6.09501456817915574557965042986, 7.00437159362150361684871153334, 7.73151981954266855813602480501, 8.677032330280436954302927106022
