L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 4·13-s + 14-s + 16-s − 2·17-s − 8·19-s + 8·23-s + 4·26-s + 28-s + 8·29-s + 4·31-s + 32-s − 2·34-s − 8·37-s − 8·38-s + 12·41-s + 8·43-s + 8·46-s − 4·47-s + 49-s + 4·52-s − 6·53-s + 56-s + 8·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.83·19-s + 1.66·23-s + 0.784·26-s + 0.188·28-s + 1.48·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s − 1.31·37-s − 1.29·38-s + 1.87·41-s + 1.21·43-s + 1.17·46-s − 0.583·47-s + 1/7·49-s + 0.554·52-s − 0.824·53-s + 0.133·56-s + 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.298015414\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.298015414\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.590177499573395911490114427595, −8.016821524264740332812865691885, −6.87277913369249633956410651955, −6.46173799242550489057344321150, −5.62569991092945877733556764079, −4.65543252751234456936091569325, −4.17626342245975600652868139578, −3.10801839200856070488758344488, −2.22534520243849483004257278768, −1.04127289295051385338945675916,
1.04127289295051385338945675916, 2.22534520243849483004257278768, 3.10801839200856070488758344488, 4.17626342245975600652868139578, 4.65543252751234456936091569325, 5.62569991092945877733556764079, 6.46173799242550489057344321150, 6.87277913369249633956410651955, 8.016821524264740332812865691885, 8.590177499573395911490114427595