L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 3·11-s + 12-s − 14-s + 16-s − 2·17-s − 18-s + 19-s + 21-s − 3·22-s + 4·23-s − 24-s + 27-s + 28-s + 8·29-s + 7·31-s − 32-s + 3·33-s + 2·34-s + 36-s + 5·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.218·21-s − 0.639·22-s + 0.834·23-s − 0.204·24-s + 0.192·27-s + 0.188·28-s + 1.48·29-s + 1.25·31-s − 0.176·32-s + 0.522·33-s + 0.342·34-s + 1/6·36-s + 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.772531922\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.772531922\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.68789182784809, −15.12077370781262, −14.69985619146052, −13.93163018044248, −13.78267689336231, −12.85197454128093, −12.29323576936311, −11.80329119840892, −11.04974743727488, −10.78982183836160, −9.837543380343194, −9.561443015875738, −8.852677283126815, −8.453740351251226, −7.875283053048419, −7.202889080620006, −6.646746970860402, −6.103203467346877, −5.195765686049119, −4.371610279655642, −3.920514631470578, −2.760489077231662, −2.524444731009534, −1.329192048663258, −0.8516034315057962,
0.8516034315057962, 1.329192048663258, 2.524444731009534, 2.760489077231662, 3.920514631470578, 4.371610279655642, 5.195765686049119, 6.103203467346877, 6.646746970860402, 7.202889080620006, 7.875283053048419, 8.453740351251226, 8.852677283126815, 9.561443015875738, 9.837543380343194, 10.78982183836160, 11.04974743727488, 11.80329119840892, 12.29323576936311, 12.85197454128093, 13.78267689336231, 13.93163018044248, 14.69985619146052, 15.12077370781262, 15.68789182784809