| L(s) = 1 | + 3-s + 3·5-s + 7-s + 9-s − 11-s − 13-s + 3·15-s + 7·17-s − 19-s + 21-s + 7·23-s + 4·25-s + 27-s + 3·29-s − 33-s + 3·35-s − 5·37-s − 39-s + 4·41-s − 11·43-s + 3·45-s + 49-s + 7·51-s − 14·53-s − 3·55-s − 57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.774·15-s + 1.69·17-s − 0.229·19-s + 0.218·21-s + 1.45·23-s + 4/5·25-s + 0.192·27-s + 0.557·29-s − 0.174·33-s + 0.507·35-s − 0.821·37-s − 0.160·39-s + 0.624·41-s − 1.67·43-s + 0.447·45-s + 1/7·49-s + 0.980·51-s − 1.92·53-s − 0.404·55-s − 0.132·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.561760821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.561760821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.297908434165637161576622367836, −7.79750031985597123022221087295, −6.87413876828884985257780712150, −6.21546419044321822792917337842, −5.18175720282501094583143650556, −4.99331147377776556304868028110, −3.56809215350041090338978969107, −2.85639683222526307064135600601, −1.95847828217916649629873361951, −1.12900213558600553009356863153,
1.12900213558600553009356863153, 1.95847828217916649629873361951, 2.85639683222526307064135600601, 3.56809215350041090338978969107, 4.99331147377776556304868028110, 5.18175720282501094583143650556, 6.21546419044321822792917337842, 6.87413876828884985257780712150, 7.79750031985597123022221087295, 8.297908434165637161576622367836
