L(s) = 1 | + 3-s − 5·7-s + 9-s + 6·11-s + 3·13-s + 2·17-s − 19-s − 5·21-s − 2·23-s + 27-s + 6·29-s − 3·31-s + 6·33-s + 6·37-s + 3·39-s + 4·41-s + 11·43-s − 10·47-s + 18·49-s + 2·51-s + 8·53-s − 57-s + 6·59-s + 3·61-s − 5·63-s − 67-s − 2·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s + 1.80·11-s + 0.832·13-s + 0.485·17-s − 0.229·19-s − 1.09·21-s − 0.417·23-s + 0.192·27-s + 1.11·29-s − 0.538·31-s + 1.04·33-s + 0.986·37-s + 0.480·39-s + 0.624·41-s + 1.67·43-s − 1.45·47-s + 18/7·49-s + 0.280·51-s + 1.09·53-s − 0.132·57-s + 0.781·59-s + 0.384·61-s − 0.629·63-s − 0.122·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.887258150\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.887258150\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 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
−9.555245357897722358921047421910, −9.128181782194024246157982259329, −8.291668725519808789628470451905, −7.08461711219266579757798770076, −6.45635118586799836731067220778, −5.86512872194008952189137540130, −4.10679719985099719204641406400, −3.65326352911551900765982300670, −2.65210489097577913822087344487, −1.05543325911168400118370111263,
1.05543325911168400118370111263, 2.65210489097577913822087344487, 3.65326352911551900765982300670, 4.10679719985099719204641406400, 5.86512872194008952189137540130, 6.45635118586799836731067220778, 7.08461711219266579757798770076, 8.291668725519808789628470451905, 9.128181782194024246157982259329, 9.555245357897722358921047421910