L(s) = 1 | + 5·7-s + 6·11-s + 3·13-s − 2·17-s + 19-s − 2·23-s − 6·29-s + 3·31-s + 6·37-s − 4·41-s − 11·43-s − 10·47-s + 18·49-s − 8·53-s + 6·59-s + 3·61-s + 67-s + 12·71-s − 10·73-s + 30·77-s − 8·79-s − 6·83-s + 16·89-s + 15·91-s + 7·97-s + 8·101-s − 4·103-s + ⋯ |
L(s) = 1 | + 1.88·7-s + 1.80·11-s + 0.832·13-s − 0.485·17-s + 0.229·19-s − 0.417·23-s − 1.11·29-s + 0.538·31-s + 0.986·37-s − 0.624·41-s − 1.67·43-s − 1.45·47-s + 18/7·49-s − 1.09·53-s + 0.781·59-s + 0.384·61-s + 0.122·67-s + 1.42·71-s − 1.17·73-s + 3.41·77-s − 0.900·79-s − 0.658·83-s + 1.69·89-s + 1.57·91-s + 0.710·97-s + 0.796·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.490534859\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.490534859\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−9.115032806662485095262867346151, −8.454615204335298608896394581701, −7.88629938412652381595519131381, −6.85475298003952077090087996844, −6.12727249083683562052212189680, −5.08671035562428751968265527166, −4.32974883216959678198797101755, −3.55986056962624680963096607393, −1.91248644348025693456689482340, −1.27192291898743446320958076704,
1.27192291898743446320958076704, 1.91248644348025693456689482340, 3.55986056962624680963096607393, 4.32974883216959678198797101755, 5.08671035562428751968265527166, 6.12727249083683562052212189680, 6.85475298003952077090087996844, 7.88629938412652381595519131381, 8.454615204335298608896394581701, 9.115032806662485095262867346151