| L(s) = 1 | + 0.540·5-s + 7-s + 1.74·11-s − 0.763·13-s + 2.28·17-s − 19-s − 7.40·23-s − 4.70·25-s − 7.94·29-s − 2.76·31-s + 0.540·35-s − 6·37-s + 1.74·41-s − 10.1·43-s − 5.78·47-s + 49-s − 6.19·53-s + 0.944·55-s + 12.6·59-s + 6.94·61-s − 0.412·65-s − 2.47·67-s + 9.69·71-s − 3.52·73-s + 1.74·77-s + 4.94·79-s + 7.27·83-s + ⋯ |
| L(s) = 1 | + 0.241·5-s + 0.377·7-s + 0.527·11-s − 0.211·13-s + 0.554·17-s − 0.229·19-s − 1.54·23-s − 0.941·25-s − 1.47·29-s − 0.496·31-s + 0.0913·35-s − 0.986·37-s + 0.273·41-s − 1.55·43-s − 0.843·47-s + 0.142·49-s − 0.851·53-s + 0.127·55-s + 1.64·59-s + 0.889·61-s − 0.0511·65-s − 0.302·67-s + 1.15·71-s − 0.412·73-s + 0.199·77-s + 0.556·79-s + 0.798·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.540T + 5T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 23 | \( 1 + 7.40T + 23T^{2} \) |
| 29 | \( 1 + 7.94T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 1.74T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 5.78T + 47T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.017103956451083334039151398849, −7.21762104642214624726940974120, −6.46666093112910879626233499270, −5.67359951482710780346531902426, −5.10838008247957123906965815884, −4.01109552123616620893476015508, −3.52814031667448356876117648730, −2.18296533398540651323550229840, −1.57463874677550758457091818522, 0,
1.57463874677550758457091818522, 2.18296533398540651323550229840, 3.52814031667448356876117648730, 4.01109552123616620893476015508, 5.10838008247957123906965815884, 5.67359951482710780346531902426, 6.46666093112910879626233499270, 7.21762104642214624726940974120, 8.017103956451083334039151398849
