| L(s) = 1 | − 1.40·3-s + 1.57·7-s − 1.03·9-s − 4.35·11-s − 0.964·13-s + 0.300·17-s + 8.62·19-s − 2.20·21-s − 23-s + 5.65·27-s + 4.76·29-s + 5.59·31-s + 6.11·33-s + 4.38·37-s + 1.35·39-s − 6.62·41-s − 1.72·43-s − 0.687·47-s − 4.53·49-s − 0.421·51-s − 8.05·53-s − 12.1·57-s − 5.74·59-s − 13.6·61-s − 1.61·63-s + 6.49·67-s + 1.40·69-s + ⋯ |
| L(s) = 1 | − 0.810·3-s + 0.593·7-s − 0.343·9-s − 1.31·11-s − 0.267·13-s + 0.0728·17-s + 1.97·19-s − 0.481·21-s − 0.208·23-s + 1.08·27-s + 0.885·29-s + 1.00·31-s + 1.06·33-s + 0.720·37-s + 0.216·39-s − 1.03·41-s − 0.263·43-s − 0.100·47-s − 0.647·49-s − 0.0589·51-s − 1.10·53-s − 1.60·57-s − 0.748·59-s − 1.74·61-s − 0.204·63-s + 0.792·67-s + 0.168·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.179322461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179322461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.40T + 3T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 + 0.964T + 13T^{2} \) |
| 17 | \( 1 - 0.300T + 17T^{2} \) |
| 19 | \( 1 - 8.62T + 19T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 4.38T + 37T^{2} \) |
| 41 | \( 1 + 6.62T + 41T^{2} \) |
| 43 | \( 1 + 1.72T + 43T^{2} \) |
| 47 | \( 1 + 0.687T + 47T^{2} \) |
| 53 | \( 1 + 8.05T + 53T^{2} \) |
| 59 | \( 1 + 5.74T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 6.49T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + 6.35T + 73T^{2} \) |
| 79 | \( 1 - 6.95T + 79T^{2} \) |
| 83 | \( 1 + 0.185T + 83T^{2} \) |
| 89 | \( 1 + 1.64T + 89T^{2} \) |
| 97 | \( 1 - 9.21T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.87561251044937135804478764106, −7.04616215580093283140604949226, −6.19296160655540979528124408319, −5.62923163077647156462500404122, −4.85704086691754657723695208883, −4.72609009512419540977989601285, −3.19896036037118686763361609116, −2.80454738554761168256656770725, −1.57722757377659723851916031626, −0.55607402985922648574029236690,
0.55607402985922648574029236690, 1.57722757377659723851916031626, 2.80454738554761168256656770725, 3.19896036037118686763361609116, 4.72609009512419540977989601285, 4.85704086691754657723695208883, 5.62923163077647156462500404122, 6.19296160655540979528124408319, 7.04616215580093283140604949226, 7.87561251044937135804478764106
