| L(s) = 1 | − 2.28·5-s − 7-s + 2.82·11-s − 3.23·13-s − 0.540·17-s + 19-s − 2.82·23-s + 0.236·25-s + 8.61·29-s + 7.70·31-s + 2.28·35-s + 4.47·37-s + 8.48·41-s − 5.23·43-s + 4.03·47-s + 49-s − 11.4·53-s − 6.47·55-s − 2.16·59-s − 3.52·61-s + 7.40·65-s − 15.4·67-s − 12.5·71-s − 10.9·73-s − 2.82·77-s − 10.4·79-s + 7.94·83-s + ⋯ |
| L(s) = 1 | − 1.02·5-s − 0.377·7-s + 0.852·11-s − 0.897·13-s − 0.131·17-s + 0.229·19-s − 0.589·23-s + 0.0472·25-s + 1.59·29-s + 1.38·31-s + 0.386·35-s + 0.735·37-s + 1.32·41-s − 0.798·43-s + 0.588·47-s + 0.142·49-s − 1.57·53-s − 0.872·55-s − 0.281·59-s − 0.451·61-s + 0.918·65-s − 1.88·67-s − 1.48·71-s − 1.28·73-s − 0.322·77-s − 1.17·79-s + 0.872·83-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 | \( 1 + 2.28T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 0.540T + 17T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 4.03T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.94T + 83T^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.83828355817243779839772732903, −7.32721028648271020291277910256, −6.46952977890732733179401067772, −5.92525980389471972325895035321, −4.52175428633810275897515521388, −4.43351066840150998649827582268, −3.29834283099864060341770962250, −2.60799913722677864600904846178, −1.21917810181995869676589755067, 0,
1.21917810181995869676589755067, 2.60799913722677864600904846178, 3.29834283099864060341770962250, 4.43351066840150998649827582268, 4.52175428633810275897515521388, 5.92525980389471972325895035321, 6.46952977890732733179401067772, 7.32721028648271020291277910256, 7.83828355817243779839772732903
