| L(s) = 1 | + 5-s − 2.20·7-s + 5.24·11-s + 13-s + 4.20·17-s + 0.952·19-s + 4.20·23-s + 25-s + 1.04·29-s + 2·31-s − 2.20·35-s − 1.15·37-s − 11.6·41-s − 4.40·43-s + 8.40·47-s − 2.15·49-s − 5.24·53-s + 5.24·55-s − 2.09·59-s + 5.15·61-s + 65-s + 10.4·67-s − 7.55·71-s − 5.35·73-s − 11.5·77-s − 5.65·79-s + 4.20·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.832·7-s + 1.58·11-s + 0.277·13-s + 1.01·17-s + 0.218·19-s + 0.876·23-s + 0.200·25-s + 0.194·29-s + 0.359·31-s − 0.372·35-s − 0.189·37-s − 1.81·41-s − 0.671·43-s + 1.22·47-s − 0.307·49-s − 0.721·53-s + 0.707·55-s − 0.272·59-s + 0.659·61-s + 0.124·65-s + 1.27·67-s − 0.896·71-s − 0.626·73-s − 1.31·77-s − 0.635·79-s + 0.455·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.306228992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.306228992\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 - 5.24T + 11T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 - 0.952T + 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.40T + 43T^{2} \) |
| 47 | \( 1 - 8.40T + 47T^{2} \) |
| 53 | \( 1 + 5.24T + 53T^{2} \) |
| 59 | \( 1 + 2.09T + 59T^{2} \) |
| 61 | \( 1 - 5.15T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 + 5.35T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−8.523725965977336421798476107478, −7.42579425744772670742948856225, −6.72504329226047226563020208140, −6.25950758886244998800979836010, −5.47087856732738884527564817489, −4.58745253258353630077914488423, −3.54674631845857609399709132959, −3.13294138621104677182467165787, −1.79139672927556100991758362601, −0.891519410460159050497903690746,
0.891519410460159050497903690746, 1.79139672927556100991758362601, 3.13294138621104677182467165787, 3.54674631845857609399709132959, 4.58745253258353630077914488423, 5.47087856732738884527564817489, 6.25950758886244998800979836010, 6.72504329226047226563020208140, 7.42579425744772670742948856225, 8.523725965977336421798476107478
