L(s) = 1 | + 4·7-s + 6·11-s + 2·13-s + 17-s + 4·19-s − 5·25-s + 4·31-s − 4·37-s − 6·41-s − 8·43-s + 9·49-s + 6·53-s − 4·61-s − 8·67-s + 2·73-s + 24·77-s − 8·79-s + 6·89-s + 8·91-s + 14·97-s − 18·101-s + 16·103-s − 6·107-s − 16·109-s + 6·113-s + 4·119-s + ⋯ |
L(s) = 1 | + 1.51·7-s + 1.80·11-s + 0.554·13-s + 0.242·17-s + 0.917·19-s − 25-s + 0.718·31-s − 0.657·37-s − 0.937·41-s − 1.21·43-s + 9/7·49-s + 0.824·53-s − 0.512·61-s − 0.977·67-s + 0.234·73-s + 2.73·77-s − 0.900·79-s + 0.635·89-s + 0.838·91-s + 1.42·97-s − 1.79·101-s + 1.57·103-s − 0.580·107-s − 1.53·109-s + 0.564·113-s + 0.366·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.595572431\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.595572431\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−8.813179685157477343661210306481, −8.269785254405673439509530684284, −7.46213856506906604015103773711, −6.65707940410074377437679235271, −5.80308270267818447243528813534, −4.94651036798605195040308986968, −4.13880943097154676314111492258, −3.35449502050879357609592921550, −1.82263022470755193054599066531, −1.19622207904654143616143844142,
1.19622207904654143616143844142, 1.82263022470755193054599066531, 3.35449502050879357609592921550, 4.13880943097154676314111492258, 4.94651036798605195040308986968, 5.80308270267818447243528813534, 6.65707940410074377437679235271, 7.46213856506906604015103773711, 8.269785254405673439509530684284, 8.813179685157477343661210306481