| L(s) = 1 | − 5-s + 0.585·7-s + 11-s + 1.41·13-s + 4.24·17-s + 1.17·19-s − 2.82·23-s + 25-s + 7.65·29-s + 4.82·31-s − 0.585·35-s − 3.65·37-s − 3.65·41-s + 9.07·43-s + 4.48·47-s − 6.65·49-s − 6.48·53-s − 55-s − 8.82·59-s + 8.82·61-s − 1.41·65-s − 8.48·67-s − 10.4·71-s + 7.07·73-s + 0.585·77-s − 0.485·79-s − 10.2·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.221·7-s + 0.301·11-s + 0.392·13-s + 1.02·17-s + 0.268·19-s − 0.589·23-s + 0.200·25-s + 1.42·29-s + 0.867·31-s − 0.0990·35-s − 0.601·37-s − 0.571·41-s + 1.38·43-s + 0.654·47-s − 0.950·49-s − 0.890·53-s − 0.134·55-s − 1.14·59-s + 1.13·61-s − 0.175·65-s − 1.03·67-s − 1.24·71-s + 0.827·73-s + 0.0667·77-s − 0.0545·79-s − 1.12·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.086456723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086456723\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 0.585T + 7T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 + 0.485T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−7.84196533970187846806394745874, −7.27455231148917980663780008306, −6.39122541649336575227585235504, −5.86797783638690144080705637099, −4.93155828520612439575850495942, −4.33628768835155389199481604448, −3.47615209050170800935456851052, −2.83488744193346577323784261823, −1.64536728341114238413282952998, −0.75366034043447184966304522189,
0.75366034043447184966304522189, 1.64536728341114238413282952998, 2.83488744193346577323784261823, 3.47615209050170800935456851052, 4.33628768835155389199481604448, 4.93155828520612439575850495942, 5.86797783638690144080705637099, 6.39122541649336575227585235504, 7.27455231148917980663780008306, 7.84196533970187846806394745874
