L(s) = 1 | + 2·7-s + 6·11-s + 4·13-s − 6·17-s + 4·19-s + 6·29-s + 4·31-s − 8·37-s + 8·43-s − 3·49-s − 6·53-s + 6·59-s + 2·61-s − 4·67-s − 12·71-s + 10·73-s + 12·77-s + 4·79-s − 12·83-s − 12·89-s + 8·91-s − 2·97-s + 6·101-s + 2·103-s − 12·107-s + 2·109-s + 6·113-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 1.80·11-s + 1.10·13-s − 1.45·17-s + 0.917·19-s + 1.11·29-s + 0.718·31-s − 1.31·37-s + 1.21·43-s − 3/7·49-s − 0.824·53-s + 0.781·59-s + 0.256·61-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 1.36·77-s + 0.450·79-s − 1.31·83-s − 1.27·89-s + 0.838·91-s − 0.203·97-s + 0.597·101-s + 0.197·103-s − 1.16·107-s + 0.191·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.540610326\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.540610326\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.722486175861551129316089497287, −7.925069985289410738500046182257, −6.85175361068861242286279094545, −6.50180716530341270499083876395, −5.59349236971546542053071949332, −4.55665350428054583925948769685, −4.05296028555192357284603965316, −3.07837259521521829219934080190, −1.79518294828466581685543030825, −1.03682851596125469877439321277,
1.03682851596125469877439321277, 1.79518294828466581685543030825, 3.07837259521521829219934080190, 4.05296028555192357284603965316, 4.55665350428054583925948769685, 5.59349236971546542053071949332, 6.50180716530341270499083876395, 6.85175361068861242286279094545, 7.925069985289410738500046182257, 8.722486175861551129316089497287