| L(s) = 1 | − 1.70·2-s + 0.903·4-s + 0.338·7-s + 1.86·8-s − 1.26·11-s − 6.57·13-s − 0.576·14-s − 4.99·16-s + 3.65·17-s + 19-s + 2.14·22-s − 5.14·23-s + 11.1·26-s + 0.305·28-s + 4.73·29-s − 3.25·31-s + 4.76·32-s − 6.23·34-s − 4.44·37-s − 1.70·38-s + 4.93·41-s − 9.32·43-s − 1.13·44-s + 8.77·46-s + 8.42·47-s − 6.88·49-s − 5.93·52-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.451·4-s + 0.127·7-s + 0.660·8-s − 0.380·11-s − 1.82·13-s − 0.154·14-s − 1.24·16-s + 0.887·17-s + 0.229·19-s + 0.458·22-s − 1.07·23-s + 2.19·26-s + 0.0577·28-s + 0.879·29-s − 0.583·31-s + 0.842·32-s − 1.06·34-s − 0.730·37-s − 0.276·38-s + 0.770·41-s − 1.42·43-s − 0.171·44-s + 1.29·46-s + 1.22·47-s − 0.983·49-s − 0.823·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6081262126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6081262126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 7 | \( 1 - 0.338T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 6.57T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 23 | \( 1 + 5.14T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 + 4.44T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 0.160T + 67T^{2} \) |
| 71 | \( 1 + 0.160T + 71T^{2} \) |
| 73 | \( 1 - 5.07T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 6.81T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.244303812458295085464086750806, −7.84363040534396751169220325909, −7.26041533638078406202889284442, −6.47010955588051229771386675098, −5.26968709310055289068508207850, −4.84707545069611946699866909884, −3.75766533530305889365928563226, −2.59345119044752587103412222480, −1.78455593641532030519857410026, −0.52287190458270869683458094291,
0.52287190458270869683458094291, 1.78455593641532030519857410026, 2.59345119044752587103412222480, 3.75766533530305889365928563226, 4.84707545069611946699866909884, 5.26968709310055289068508207850, 6.47010955588051229771386675098, 7.26041533638078406202889284442, 7.84363040534396751169220325909, 8.244303812458295085464086750806
