L(s) = 1 | + 2·3-s − 7-s + 9-s + 3·11-s + 4·13-s + 2·19-s − 2·21-s + 3·23-s − 4·27-s + 9·29-s + 8·31-s + 6·33-s − 5·37-s + 8·39-s − 6·41-s − 11·43-s − 6·47-s + 49-s − 6·53-s + 4·57-s − 10·61-s − 63-s − 5·67-s + 6·69-s + 15·71-s + 10·73-s − 3·77-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 0.458·19-s − 0.436·21-s + 0.625·23-s − 0.769·27-s + 1.67·29-s + 1.43·31-s + 1.04·33-s − 0.821·37-s + 1.28·39-s − 0.937·41-s − 1.67·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.125·63-s − 0.610·67-s + 0.722·69-s + 1.78·71-s + 1.17·73-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.273953882\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.273953882\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−10.21166762830498566503871071366, −9.476594992197793863667451207273, −8.579217616319689592905261623702, −8.230425389995018164891979635721, −6.89471844750543120668096034614, −6.23170776253627664590581793436, −4.81132951055676796679264238406, −3.58632498964805214399996471224, −2.96248931326365909517978646582, −1.43028021441989809127723450150,
1.43028021441989809127723450150, 2.96248931326365909517978646582, 3.58632498964805214399996471224, 4.81132951055676796679264238406, 6.23170776253627664590581793436, 6.89471844750543120668096034614, 8.230425389995018164891979635721, 8.579217616319689592905261623702, 9.476594992197793863667451207273, 10.21166762830498566503871071366