L(s) = 1 | + 3-s + 4·7-s − 2·9-s − 3·11-s − 2·13-s − 6·17-s + 4·21-s + 6·23-s − 5·25-s − 5·27-s + 2·31-s − 3·33-s + 10·37-s − 2·39-s − 9·41-s + 4·43-s + 9·49-s − 6·51-s − 6·53-s − 9·59-s − 4·61-s − 8·63-s − 7·67-s + 6·69-s − 6·71-s − 73-s − 5·75-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.904·11-s − 0.554·13-s − 1.45·17-s + 0.872·21-s + 1.25·23-s − 25-s − 0.962·27-s + 0.359·31-s − 0.522·33-s + 1.64·37-s − 0.320·39-s − 1.40·41-s + 0.609·43-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 1.17·59-s − 0.512·61-s − 1.00·63-s − 0.855·67-s + 0.722·69-s − 0.712·71-s − 0.117·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.81761069081087777970322499146, −7.36248839682354505915033410499, −6.31158630120358497541295468271, −5.46327518768877699249188244534, −4.78861690653198140987181642111, −4.26022238479431323713239583799, −2.97265058742374240434054691416, −2.42116921482369257990703314229, −1.56156670165947028241928151254, 0,
1.56156670165947028241928151254, 2.42116921482369257990703314229, 2.97265058742374240434054691416, 4.26022238479431323713239583799, 4.78861690653198140987181642111, 5.46327518768877699249188244534, 6.31158630120358497541295468271, 7.36248839682354505915033410499, 7.81761069081087777970322499146