L(s) = 1 | + 5-s + 4·7-s + 2·13-s − 6·17-s + 4·19-s + 25-s + 6·29-s − 8·31-s + 4·35-s + 2·37-s + 6·41-s + 4·43-s + 9·49-s + 6·53-s − 10·61-s + 2·65-s + 4·67-s + 2·73-s − 8·79-s + 12·83-s − 6·85-s − 18·89-s + 8·91-s + 4·95-s + 2·97-s − 18·101-s + 4·103-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.824·53-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.234·73-s − 0.900·79-s + 1.31·83-s − 0.650·85-s − 1.90·89-s + 0.838·91-s + 0.410·95-s + 0.203·97-s − 1.79·101-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.935248229\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.935248229\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.67015888052155601125139797767, −9.415401762585890013045544643092, −8.713350404322888410914105544855, −7.88297907167905508023143292047, −6.96170559755127999142057961261, −5.85236328773314391596499497567, −4.97026786036012165020386374880, −4.08471431361169401081275954745, −2.50374962358163435732612030839, −1.36243712364701817507861978801,
1.36243712364701817507861978801, 2.50374962358163435732612030839, 4.08471431361169401081275954745, 4.97026786036012165020386374880, 5.85236328773314391596499497567, 6.96170559755127999142057961261, 7.88297907167905508023143292047, 8.713350404322888410914105544855, 9.415401762585890013045544643092, 10.67015888052155601125139797767