L(s) = 1 | − 2·3-s + 9-s − 11-s − 2·13-s + 4·17-s − 5·23-s + 4·27-s − 3·29-s + 10·31-s + 2·33-s − 5·37-s + 4·39-s + 10·41-s + 5·43-s − 4·47-s − 8·51-s − 10·53-s + 10·59-s − 10·61-s − 5·67-s + 10·69-s + 3·71-s − 10·73-s + 13·79-s − 11·81-s − 10·83-s + 6·87-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.970·17-s − 1.04·23-s + 0.769·27-s − 0.557·29-s + 1.79·31-s + 0.348·33-s − 0.821·37-s + 0.640·39-s + 1.56·41-s + 0.762·43-s − 0.583·47-s − 1.12·51-s − 1.37·53-s + 1.30·59-s − 1.28·61-s − 0.610·67-s + 1.20·69-s + 0.356·71-s − 1.17·73-s + 1.46·79-s − 1.22·81-s − 1.09·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.79668235407517820657938266629, −7.17237222162414252035660761333, −6.15079025199181072961667129444, −5.88537832785510657585344947547, −4.99888912676944807075280885451, −4.42722524354363595500927984615, −3.32290859529051361238778095103, −2.37876453878142867430885314451, −1.11352695600890301200390605922, 0,
1.11352695600890301200390605922, 2.37876453878142867430885314451, 3.32290859529051361238778095103, 4.42722524354363595500927984615, 4.99888912676944807075280885451, 5.88537832785510657585344947547, 6.15079025199181072961667129444, 7.17237222162414252035660761333, 7.79668235407517820657938266629