L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s − 5·13-s − 2·14-s + 15-s + 16-s − 3·17-s − 18-s − 19-s − 20-s − 2·21-s − 4·22-s + 24-s + 25-s + 5·26-s − 27-s + 2·28-s + 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 1.38·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.436·21-s − 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.192·27-s + 0.377·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5712737618\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5712737618\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−12.28060792183461, −12.19908025170277, −11.67936456653588, −11.24766864004176, −11.09373973298519, −10.21749292780123, −10.03782083993181, −9.549374175393298, −8.939652656402692, −8.476096174796926, −8.180086046466444, −7.556441564616523, −6.921125071852661, −6.817799009538872, −6.359328683541265, −5.425032634242641, −5.205089153909806, −4.560267779255234, −4.092006377492083, −3.622102368404098, −2.698704183508167, −2.281982591933631, −1.553587550062551, −1.108649510298491, −0.2537948520717381,
0.2537948520717381, 1.108649510298491, 1.553587550062551, 2.281982591933631, 2.698704183508167, 3.622102368404098, 4.092006377492083, 4.560267779255234, 5.205089153909806, 5.425032634242641, 6.359328683541265, 6.817799009538872, 6.921125071852661, 7.556441564616523, 8.180086046466444, 8.476096174796926, 8.939652656402692, 9.549374175393298, 10.03782083993181, 10.21749292780123, 11.09373973298519, 11.24766864004176, 11.67936456653588, 12.19908025170277, 12.28060792183461