L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 1.73·11-s + 5.46·13-s + 14-s + 16-s − 4·17-s + 5.73·19-s + 1.73·22-s − 2.46·23-s − 5.46·26-s − 28-s − 1.46·29-s − 0.267·31-s − 32-s + 4·34-s − 3.19·37-s − 5.73·38-s + 3.92·41-s − 4.53·43-s − 1.73·44-s + 2.46·46-s − 11.4·47-s + 49-s + 5.46·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.377·7-s − 0.353·8-s − 0.522·11-s + 1.51·13-s + 0.267·14-s + 0.250·16-s − 0.970·17-s + 1.31·19-s + 0.369·22-s − 0.513·23-s − 1.07·26-s − 0.188·28-s − 0.271·29-s − 0.0481·31-s − 0.176·32-s + 0.685·34-s − 0.525·37-s − 0.929·38-s + 0.613·41-s − 0.691·43-s − 0.261·44-s + 0.363·46-s − 1.67·47-s + 0.142·49-s + 0.757·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + 0.267T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.26T + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 - 8.92T + 79T^{2} \) |
| 83 | \( 1 + 9.46T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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
−7.43981934414089453717932559479, −6.73591336736387962480539343883, −6.10170428938133364198218391723, −5.50481074554417891210822603839, −4.56538579392919930440680499187, −3.60849592097197055350389024152, −3.04445184848949893394540452143, −2.01847819875614829427179538338, −1.15658120389401157436007969608, 0,
1.15658120389401157436007969608, 2.01847819875614829427179538338, 3.04445184848949893394540452143, 3.60849592097197055350389024152, 4.56538579392919930440680499187, 5.50481074554417891210822603839, 6.10170428938133364198218391723, 6.73591336736387962480539343883, 7.43981934414089453717932559479