| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 8-s + 9-s + 3·10-s + 3·11-s − 12-s − 5·13-s − 3·15-s + 16-s + 17-s + 18-s − 2·19-s + 3·20-s + 3·22-s + 6·23-s − 24-s + 4·25-s − 5·26-s − 27-s − 6·29-s − 3·30-s + 4·31-s + 32-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 + 0.904·11-s − 0.288·12-s − 1.38·13-s − 0.774·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.670·20-s + 0.639·22-s + 1.25·23-s − 0.204·24-s + 4/5·25-s − 0.980·26-s − 0.192·27-s − 1.11·29-s − 0.547·30-s + 0.718·31-s + 0.176·32-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
| \(L(1)\) |
\(\approx\) |
\(3.544394140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.544394140\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−8.083165555248592147429688098051, −7.23094670024759004372563807504, −6.53957843456583678828702980252, −6.08141158376882457110354015136, −5.27519380144117488220199487352, −4.78546933525726930967611120365, −3.89603433535508610703560974588, −2.72761250465654495662053852708, −2.05153640888965011495979218732, −0.991619896358621361811920956660,
0.991619896358621361811920956660, 2.05153640888965011495979218732, 2.72761250465654495662053852708, 3.89603433535508610703560974588, 4.78546933525726930967611120365, 5.27519380144117488220199487352, 6.08141158376882457110354015136, 6.53957843456583678828702980252, 7.23094670024759004372563807504, 8.083165555248592147429688098051
