L(s) = 1 | + 2·7-s − 11-s − 6·13-s − 6·17-s + 2·19-s − 8·23-s − 5·25-s + 2·29-s + 4·31-s + 2·37-s − 10·41-s + 6·43-s + 4·47-s − 3·49-s + 4·53-s − 4·59-s − 2·61-s + 8·67-s − 12·71-s − 2·73-s − 2·77-s − 14·79-s + 4·83-s − 12·91-s + 2·97-s − 14·101-s + 16·103-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 0.301·11-s − 1.66·13-s − 1.45·17-s + 0.458·19-s − 1.66·23-s − 25-s + 0.371·29-s + 0.718·31-s + 0.328·37-s − 1.56·41-s + 0.914·43-s + 0.583·47-s − 3/7·49-s + 0.549·53-s − 0.520·59-s − 0.256·61-s + 0.977·67-s − 1.42·71-s − 0.234·73-s − 0.227·77-s − 1.57·79-s + 0.439·83-s − 1.25·91-s + 0.203·97-s − 1.39·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−9.017507074484416772068251654581, −8.099020164385146094336344333963, −7.56268069174475458172560677764, −6.66074445750953603088024612408, −5.66400444152815488371172603048, −4.77072318914536653364587076952, −4.14662801832066552357258286543, −2.66313192224716466246405612340, −1.88127298605493063337695627679, 0,
1.88127298605493063337695627679, 2.66313192224716466246405612340, 4.14662801832066552357258286543, 4.77072318914536653364587076952, 5.66400444152815488371172603048, 6.66074445750953603088024612408, 7.56268069174475458172560677764, 8.099020164385146094336344333963, 9.017507074484416772068251654581