L(s) = 1 | + 5-s − 2·7-s − 4·11-s + 2·13-s − 5·17-s − 5·19-s + 23-s + 25-s − 2·29-s − 7·31-s − 2·35-s + 6·37-s + 4·43-s + 4·47-s − 3·49-s + 9·53-s − 4·55-s − 14·59-s + 11·61-s + 2·65-s + 14·67-s − 12·73-s + 8·77-s + 3·79-s + 83-s − 5·85-s − 4·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.554·13-s − 1.21·17-s − 1.14·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 1.25·31-s − 0.338·35-s + 0.986·37-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 1.23·53-s − 0.539·55-s − 1.82·59-s + 1.40·61-s + 0.248·65-s + 1.71·67-s − 1.40·73-s + 0.911·77-s + 0.337·79-s + 0.109·83-s − 0.542·85-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.243115331\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.243115331\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.69381804337674001973337841786, −7.06666940321106711649016532162, −6.27971706861172856969557550419, −5.86740150416520331510513417666, −5.02277539706067172285035752952, −4.25072470920032214502565100153, −3.44931407523417704011114555728, −2.51930996021286779208892591113, −1.98338657662642604152391065145, −0.51275164904465627438494634049,
0.51275164904465627438494634049, 1.98338657662642604152391065145, 2.51930996021286779208892591113, 3.44931407523417704011114555728, 4.25072470920032214502565100153, 5.02277539706067172285035752952, 5.86740150416520331510513417666, 6.27971706861172856969557550419, 7.06666940321106711649016532162, 7.69381804337674001973337841786