L(s) = 1 | + 7-s − 3·9-s − 4·13-s − 4·17-s − 4·19-s + 8·23-s + 2·29-s + 8·31-s + 8·37-s + 6·41-s + 8·43-s + 8·47-s + 49-s + 4·59-s − 6·61-s − 3·63-s + 8·67-s − 12·71-s + 4·73-s + 4·79-s + 9·81-s − 10·89-s − 4·91-s + 12·97-s − 18·101-s − 8·103-s + 8·107-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 9-s − 1.10·13-s − 0.970·17-s − 0.917·19-s + 1.66·23-s + 0.371·29-s + 1.43·31-s + 1.31·37-s + 0.937·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.520·59-s − 0.768·61-s − 0.377·63-s + 0.977·67-s − 1.42·71-s + 0.468·73-s + 0.450·79-s + 81-s − 1.05·89-s − 0.419·91-s + 1.21·97-s − 1.79·101-s − 0.788·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.507427941\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.507427941\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.842473363162386067702191588923, −8.100019357488912466620887602522, −7.31620836995693460320439771002, −6.51061195379737176118295534343, −5.74173198413028645208184945723, −4.79990698099335600777816507759, −4.26879354614828383723940391147, −2.81078741791563822701964316606, −2.38148268527460349076544066038, −0.74077423986771590733306844987,
0.74077423986771590733306844987, 2.38148268527460349076544066038, 2.81078741791563822701964316606, 4.26879354614828383723940391147, 4.79990698099335600777816507759, 5.74173198413028645208184945723, 6.51061195379737176118295534343, 7.31620836995693460320439771002, 8.100019357488912466620887602522, 8.842473363162386067702191588923