L(s) = 1 | − 3·7-s + 2·11-s − 13-s + 2·17-s + 5·19-s − 6·23-s − 10·29-s + 3·31-s − 2·37-s + 8·41-s + 43-s − 2·47-s + 2·49-s − 4·53-s − 10·59-s + 7·61-s − 3·67-s − 8·71-s + 14·73-s − 6·77-s − 6·83-s + 3·91-s − 17·97-s − 12·101-s − 4·103-s − 12·107-s + 5·109-s + ⋯ |
L(s) = 1 | − 1.13·7-s + 0.603·11-s − 0.277·13-s + 0.485·17-s + 1.14·19-s − 1.25·23-s − 1.85·29-s + 0.538·31-s − 0.328·37-s + 1.24·41-s + 0.152·43-s − 0.291·47-s + 2/7·49-s − 0.549·53-s − 1.30·59-s + 0.896·61-s − 0.366·67-s − 0.949·71-s + 1.63·73-s − 0.683·77-s − 0.658·83-s + 0.314·91-s − 1.72·97-s − 1.19·101-s − 0.394·103-s − 1.16·107-s + 0.478·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.045639809832312220968668237407, −7.46678658042041033049343800975, −6.66175602749173699922625283942, −5.96391714742476427877734512505, −5.31508624669203565794999838972, −4.12503217255581132297434015747, −3.50877292142339574199218689538, −2.63519340777820009043746148182, −1.40715345955998479729965074286, 0,
1.40715345955998479729965074286, 2.63519340777820009043746148182, 3.50877292142339574199218689538, 4.12503217255581132297434015747, 5.31508624669203565794999838972, 5.96391714742476427877734512505, 6.66175602749173699922625283942, 7.46678658042041033049343800975, 8.045639809832312220968668237407