L(s) = 1 | + 2·5-s + 6·13-s − 2·17-s − 25-s + 10·29-s − 2·37-s − 10·41-s − 7·49-s − 14·53-s − 10·61-s + 12·65-s − 6·73-s − 4·85-s − 10·89-s + 18·97-s + 2·101-s + 6·109-s + 14·113-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.66·13-s − 0.485·17-s − 1/5·25-s + 1.85·29-s − 0.328·37-s − 1.56·41-s − 49-s − 1.92·53-s − 1.28·61-s + 1.48·65-s − 0.702·73-s − 0.433·85-s − 1.05·89-s + 1.82·97-s + 0.199·101-s + 0.574·109-s + 1.31·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.513845634\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.513845634\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−11.74104183190432762188729414275, −10.79424580459691548950831956254, −9.988093934240586032989036843847, −8.944850116469648867972974935175, −8.155593376401658883389001410970, −6.62912355829218159331214255175, −5.98788019309426758968327152208, −4.69735185642958092576120623346, −3.23795791772265289886094041940, −1.61821392697617915675557729976,
1.61821392697617915675557729976, 3.23795791772265289886094041940, 4.69735185642958092576120623346, 5.98788019309426758968327152208, 6.62912355829218159331214255175, 8.155593376401658883389001410970, 8.944850116469648867972974935175, 9.988093934240586032989036843847, 10.79424580459691548950831956254, 11.74104183190432762188729414275