| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 8-s + 9-s − 3·10-s + 11-s + 12-s − 13-s − 3·15-s + 16-s + 17-s + 18-s − 6·19-s − 3·20-s + 22-s − 2·23-s + 24-s + 4·25-s − 26-s + 27-s − 2·29-s − 3·30-s + 32-s + 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.774·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.37·19-s − 0.670·20-s + 0.213·22-s − 0.417·23-s + 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.371·29-s − 0.547·30-s + 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−7.85059617618581656779373839144, −7.20882905417416568939645115117, −6.53915924588061758306737756183, −5.65847197016302973824328258878, −4.56674591672556234699502232416, −4.16593315979626522408032719166, −3.45831093269106852096481812768, −2.65658039843883482896578852490, −1.58963238181820946376509190337, 0,
1.58963238181820946376509190337, 2.65658039843883482896578852490, 3.45831093269106852096481812768, 4.16593315979626522408032719166, 4.56674591672556234699502232416, 5.65847197016302973824328258878, 6.53915924588061758306737756183, 7.20882905417416568939645115117, 7.85059617618581656779373839144
