L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 6·11-s + 12-s + 6·13-s + 15-s + 16-s − 18-s − 4·19-s + 20-s − 6·22-s − 24-s + 25-s − 6·26-s + 27-s − 8·29-s − 30-s + 2·31-s − 32-s + 6·33-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s + 0.288·12-s + 1.66·13-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.27·22-s − 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s − 1.48·29-s − 0.182·30-s + 0.359·31-s − 0.176·32-s + 1.04·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.930389907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.930389907\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.460411793280627593776904848262, −8.521238822857409210934995529736, −8.434314307117336498034641568137, −6.95296994643615766643683231422, −6.52737336543834537369197277554, −5.63274955765687972234201249790, −4.08749373285089222633534816883, −3.48751416538174277232886774765, −2.03281451774888971137165966691, −1.21094607998653189520756553865,
1.21094607998653189520756553865, 2.03281451774888971137165966691, 3.48751416538174277232886774765, 4.08749373285089222633534816883, 5.63274955765687972234201249790, 6.52737336543834537369197277554, 6.95296994643615766643683231422, 8.434314307117336498034641568137, 8.521238822857409210934995529736, 9.460411793280627593776904848262