| L(s) = 1 | − 2.38·2-s + 3.69·4-s + 2.27·5-s + 2.95·7-s − 4.03·8-s − 5.42·10-s − 11-s + 7.16·13-s − 7.04·14-s + 2.24·16-s − 3.73·17-s − 19-s + 8.39·20-s + 2.38·22-s + 7.03·23-s + 0.175·25-s − 17.0·26-s + 10.9·28-s − 0.694·29-s + 7.76·31-s + 2.72·32-s + 8.91·34-s + 6.71·35-s + 0.351·37-s + 2.38·38-s − 9.17·40-s + 6.87·41-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.84·4-s + 1.01·5-s + 1.11·7-s − 1.42·8-s − 1.71·10-s − 0.301·11-s + 1.98·13-s − 1.88·14-s + 0.560·16-s − 0.906·17-s − 0.229·19-s + 1.87·20-s + 0.508·22-s + 1.46·23-s + 0.0350·25-s − 3.35·26-s + 2.06·28-s − 0.128·29-s + 1.39·31-s + 0.481·32-s + 1.52·34-s + 1.13·35-s + 0.0578·37-s + 0.386·38-s − 1.45·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.207864978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207864978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 - 2.95T + 7T^{2} \) |
| 13 | \( 1 - 7.16T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 23 | \( 1 - 7.03T + 23T^{2} \) |
| 29 | \( 1 + 0.694T + 29T^{2} \) |
| 31 | \( 1 - 7.76T + 31T^{2} \) |
| 37 | \( 1 - 0.351T + 37T^{2} \) |
| 41 | \( 1 - 6.87T + 41T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 - 1.84T + 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 + 4.78T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.60T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 2.47T + 97T^{2} \) |
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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
−9.088734423470266066898991360478, −8.497996215662819403787143743366, −8.097722564062521266846152955460, −6.96819729760627841041264638939, −6.33833807422506846072972386020, −5.45285730199311404092744559069, −4.32464947405083307117047804410, −2.75738606726511648373571044488, −1.75934227434056692752838724584, −1.05993124218341352240536160595,
1.05993124218341352240536160595, 1.75934227434056692752838724584, 2.75738606726511648373571044488, 4.32464947405083307117047804410, 5.45285730199311404092744559069, 6.33833807422506846072972386020, 6.96819729760627841041264638939, 8.097722564062521266846152955460, 8.497996215662819403787143743366, 9.088734423470266066898991360478
