| L(s) = 1 | − 5-s − 0.335·7-s − 1.19·11-s + 3.77·13-s + 4.01·17-s − 4.01·19-s − 6.81·23-s + 25-s − 1.90·29-s + 7.48·31-s + 0.335·35-s − 37-s − 11.9·41-s + 2.09·43-s − 0.495·47-s − 6.88·49-s + 1.71·53-s + 1.19·55-s + 8.06·59-s − 5.21·61-s − 3.77·65-s + 6.68·67-s − 0.254·71-s − 8.16·73-s + 0.402·77-s + 17.0·79-s − 6.03·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.126·7-s − 0.361·11-s + 1.04·13-s + 0.974·17-s − 0.921·19-s − 1.42·23-s + 0.200·25-s − 0.353·29-s + 1.34·31-s + 0.0566·35-s − 0.164·37-s − 1.86·41-s + 0.319·43-s − 0.0723·47-s − 0.983·49-s + 0.235·53-s + 0.161·55-s + 1.05·59-s − 0.667·61-s − 0.468·65-s + 0.817·67-s − 0.0301·71-s − 0.955·73-s + 0.0458·77-s + 1.91·79-s − 0.661·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 0.335T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 19 | \( 1 + 4.01T + 19T^{2} \) |
| 23 | \( 1 + 6.81T + 23T^{2} \) |
| 29 | \( 1 + 1.90T + 29T^{2} \) |
| 31 | \( 1 - 7.48T + 31T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 + 0.495T + 47T^{2} \) |
| 53 | \( 1 - 1.71T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 + 5.21T + 61T^{2} \) |
| 67 | \( 1 - 6.68T + 67T^{2} \) |
| 71 | \( 1 + 0.254T + 71T^{2} \) |
| 73 | \( 1 + 8.16T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 6.03T + 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.916729815967228238013266395722, −6.84292122986857228079790279750, −6.27957532103197426516195580191, −5.57512108988209610010815164772, −4.72696153881583933403138893617, −3.88001760134216041314127893965, −3.34748477917802424136856471130, −2.29464008246686050305879212392, −1.26792086399335809550755045083, 0,
1.26792086399335809550755045083, 2.29464008246686050305879212392, 3.34748477917802424136856471130, 3.88001760134216041314127893965, 4.72696153881583933403138893617, 5.57512108988209610010815164772, 6.27957532103197426516195580191, 6.84292122986857228079790279750, 7.916729815967228238013266395722
