| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s − 2·7-s + 8·8-s + 9·9-s − 12·12-s − 13·13-s − 4·14-s + 16·16-s + 60·17-s + 18·18-s + 50·19-s + 6·21-s − 210·23-s − 24·24-s − 26·26-s − 27·27-s − 8·28-s − 228·29-s + 116·31-s + 32·32-s + 120·34-s + 36·36-s − 386·37-s + 100·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.107·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.277·13-s − 0.0763·14-s + 1/4·16-s + 0.856·17-s + 0.235·18-s + 0.603·19-s + 0.0623·21-s − 1.90·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.0539·28-s − 1.45·29-s + 0.672·31-s + 0.176·32-s + 0.605·34-s + 1/6·36-s − 1.71·37-s + 0.426·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 - 60 T + p^{3} T^{2} \) |
| 19 | \( 1 - 50 T + p^{3} T^{2} \) |
| 23 | \( 1 + 210 T + p^{3} T^{2} \) |
| 29 | \( 1 + 228 T + p^{3} T^{2} \) |
| 31 | \( 1 - 116 T + p^{3} T^{2} \) |
| 37 | \( 1 + 386 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 312 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 624 T + p^{3} T^{2} \) |
| 61 | \( 1 - 638 T + p^{3} T^{2} \) |
| 67 | \( 1 + 200 T + p^{3} T^{2} \) |
| 71 | \( 1 + 408 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1148 T + p^{3} T^{2} \) |
| 79 | \( 1 - 824 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1332 T + p^{3} T^{2} \) |
| 89 | \( 1 - 54 T + p^{3} T^{2} \) |
| 97 | \( 1 - 244 T + p^{3} T^{2} \) |
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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
−8.227242464152285889371627381667, −7.48088402585942437825956767211, −6.77409466027208842368518720973, −5.69690865065982669371316510038, −5.48852757114580350590433392295, −4.27904075940193801244789926575, −3.62714886672058298986895983690, −2.45670193799714750053080382221, −1.36219117323280163288078556190, 0,
1.36219117323280163288078556190, 2.45670193799714750053080382221, 3.62714886672058298986895983690, 4.27904075940193801244789926575, 5.48852757114580350590433392295, 5.69690865065982669371316510038, 6.77409466027208842368518720973, 7.48088402585942437825956767211, 8.227242464152285889371627381667
