L(s) = 1 | + 2·3-s + 9-s − 3·11-s + 4·13-s + 2·19-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 + 6·53-s + 4·57-s − 10·61-s + 5·67-s + 6·69-s + 15·71-s − 10·73-s − 7·79-s − 11·81-s − 12·83-s − 18·87-s + 12·89-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.458·19-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 + 0.824·53-s + 0.529·57-s − 1.28·61-s + 0.610·67-s + 0.722·69-s + 1.78·71-s − 1.17·73-s − 0.787·79-s − 1.22·81-s − 1.31·83-s − 1.92·87-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.495109343\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.495109343\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
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 | \( 1 - 8 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.07724391969968, −13.36285845949055, −13.24149999812608, −12.75994360058095, −12.15464104446713, −11.28747681019756, −11.01824910342120, −10.66009936826860, −9.808012034101810, −9.326672485036144, −8.957696777501012, −8.565617439474826, −7.756262797494924, −7.587606816290802, −7.119399340342508, −6.114262860921278, −5.698189522975306, −5.279145565960902, −4.305953034109945, −3.835088759700128, −3.325087444026506, −2.673895478808417, −2.201718130072384, −1.439852800372830, −0.5610645006400046,
0.5610645006400046, 1.439852800372830, 2.201718130072384, 2.673895478808417, 3.325087444026506, 3.835088759700128, 4.305953034109945, 5.279145565960902, 5.698189522975306, 6.114262860921278, 7.119399340342508, 7.587606816290802, 7.756262797494924, 8.565617439474826, 8.957696777501012, 9.326672485036144, 9.808012034101810, 10.66009936826860, 11.01824910342120, 11.28747681019756, 12.15464104446713, 12.75994360058095, 13.24149999812608, 13.36285845949055, 14.07724391969968