| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s − 3.73·11-s + 3.46·13-s + 2·15-s − 21-s − 2.53·23-s − 25-s + 27-s − 3.46·29-s − 9.46·31-s − 3.73·33-s − 2·35-s − 5.92·37-s + 3.46·39-s − 11.1·41-s + 3.46·43-s + 2·45-s − 0.267·47-s − 6·49-s + 4.26·53-s − 7.46·55-s + 6.66·59-s − 7.46·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 0.333·9-s − 1.12·11-s + 0.960·13-s + 0.516·15-s − 0.218·21-s − 0.528·23-s − 0.200·25-s + 0.192·27-s − 0.643·29-s − 1.69·31-s − 0.649·33-s − 0.338·35-s − 0.974·37-s + 0.554·39-s − 1.74·41-s + 0.528·43-s + 0.298·45-s − 0.0390·47-s − 0.857·49-s + 0.586·53-s − 1.00·55-s + 0.867·59-s − 0.955·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 + 5.92T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 0.267T + 47T^{2} \) |
| 53 | \( 1 - 4.26T + 53T^{2} \) |
| 59 | \( 1 - 6.66T + 59T^{2} \) |
| 61 | \( 1 + 7.46T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 - 5.92T + 97T^{2} \) |
| 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
−7.58199417046576141442473599520, −6.95266507794169299231172195954, −6.01928793397062664150670853988, −5.61881525253837903799018589235, −4.79659536559200984194091609318, −3.70890945026565237236807520339, −3.19544935315973779246484097883, −2.15756641418564640731210117539, −1.60901579151878517292958857388, 0,
1.60901579151878517292958857388, 2.15756641418564640731210117539, 3.19544935315973779246484097883, 3.70890945026565237236807520339, 4.79659536559200984194091609318, 5.61881525253837903799018589235, 6.01928793397062664150670853988, 6.95266507794169299231172195954, 7.58199417046576141442473599520
