| L(s) = 1 | − 2·3-s − 2·5-s − 7-s + 9-s + 2·13-s + 4·15-s + 6·17-s − 19-s + 2·21-s − 25-s + 4·27-s + 8·29-s + 2·35-s − 2·37-s − 4·39-s + 2·41-s + 4·43-s − 2·45-s − 47-s + 49-s − 12·51-s − 6·53-s + 2·57-s + 14·59-s − 6·61-s − 63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.03·15-s + 1.45·17-s − 0.229·19-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 1.48·29-s + 0.338·35-s − 0.328·37-s − 0.640·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 0.145·47-s + 1/7·49-s − 1.68·51-s − 0.824·53-s + 0.264·57-s + 1.82·59-s − 0.768·61-s − 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.97704811659913, −13.52908241774755, −12.72380856632553, −12.42419147220131, −11.99227569473500, −11.64575169866439, −11.13720472561894, −10.56171083529461, −10.32335131176209, −9.618952267991816, −9.108495446077465, −8.305528685436249, −8.053909265587366, −7.495260342648930, −6.728886178342144, −6.459331290176532, −5.842320318100381, −5.340339539698338, −4.858523530734002, −4.142737444151854, −3.651130647195894, −3.089922677536598, −2.366680154717100, −1.241794122781707, −0.7820634336751139, 0,
0.7820634336751139, 1.241794122781707, 2.366680154717100, 3.089922677536598, 3.651130647195894, 4.142737444151854, 4.858523530734002, 5.340339539698338, 5.842320318100381, 6.459331290176532, 6.728886178342144, 7.495260342648930, 8.053909265587366, 8.305528685436249, 9.108495446077465, 9.618952267991816, 10.32335131176209, 10.56171083529461, 11.13720472561894, 11.64575169866439, 11.99227569473500, 12.42419147220131, 12.72380856632553, 13.52908241774755, 13.97704811659913
