| L(s) = 1 | − 1.42·5-s − 7-s + 3.27·11-s + 3.42·13-s − 5.27·17-s − 19-s − 0.574·23-s − 2.96·25-s + 0.122·29-s + 2.12·31-s + 1.42·35-s − 3.96·37-s + 4.66·41-s − 11.9·43-s − 3.11·47-s + 49-s + 6.08·53-s − 4.66·55-s + 1.72·59-s + 12.2·61-s − 4.88·65-s − 5.27·67-s + 6.81·71-s − 11.5·73-s − 3.27·77-s + 11.0·79-s − 5.23·83-s + ⋯ |
| L(s) = 1 | − 0.637·5-s − 0.377·7-s + 0.986·11-s + 0.950·13-s − 1.27·17-s − 0.229·19-s − 0.119·23-s − 0.593·25-s + 0.0227·29-s + 0.381·31-s + 0.241·35-s − 0.652·37-s + 0.728·41-s − 1.81·43-s − 0.454·47-s + 0.142·49-s + 0.836·53-s − 0.628·55-s + 0.224·59-s + 1.56·61-s − 0.605·65-s − 0.643·67-s + 0.809·71-s − 1.35·73-s − 0.372·77-s + 1.24·79-s − 0.574·83-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 | \( 1 + 1.42T + 5T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 - 3.42T + 13T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 23 | \( 1 + 0.574T + 23T^{2} \) |
| 29 | \( 1 - 0.122T + 29T^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 3.11T + 47T^{2} \) |
| 53 | \( 1 - 6.08T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 - 6.81T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 5.23T + 83T^{2} \) |
| 89 | \( 1 - 1.45T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−8.082853725293927203213709871720, −7.01460571540713490223087234466, −6.58981154981396501174413316534, −5.89121501983364901012169717262, −4.84329458615620407145141925152, −3.95715600855877940973954997520, −3.61417168403294593911068923119, −2.40920811542236548713162993931, −1.32804388299010437186613276852, 0,
1.32804388299010437186613276852, 2.40920811542236548713162993931, 3.61417168403294593911068923119, 3.95715600855877940973954997520, 4.84329458615620407145141925152, 5.89121501983364901012169717262, 6.58981154981396501174413316534, 7.01460571540713490223087234466, 8.082853725293927203213709871720
