L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3·7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 13-s − 3·14-s + 15-s + 16-s + 2·17-s − 18-s + 20-s + 3·21-s + 22-s + 5·23-s − 24-s + 25-s + 26-s + 27-s + 3·28-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.654·21-s + 0.213·22-s + 1.04·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.566·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.538314351\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.538314351\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.62342118073906, −12.07794781151522, −11.59195770343977, −11.15209000049312, −10.67816671944314, −10.40320568398482, −9.650162767687217, −9.490471631299580, −8.942757973927004, −8.351041610819504, −8.199142084224064, −7.540935087445020, −7.238577208026637, −6.767232580016213, −6.065657999306947, −5.458545968202425, −5.215332718506591, −4.449398182429044, −4.106492386458492, −3.188867432050278, −2.830361573529029, −2.249659724933987, −1.673791749950038, −1.225710813653860, −0.5456433767813105,
0.5456433767813105, 1.225710813653860, 1.673791749950038, 2.249659724933987, 2.830361573529029, 3.188867432050278, 4.106492386458492, 4.449398182429044, 5.215332718506591, 5.458545968202425, 6.065657999306947, 6.767232580016213, 7.238577208026637, 7.540935087445020, 8.199142084224064, 8.351041610819504, 8.942757973927004, 9.490471631299580, 9.650162767687217, 10.40320568398482, 10.67816671944314, 11.15209000049312, 11.59195770343977, 12.07794781151522, 12.62342118073906