L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 4.74·13-s + 14-s + 16-s + 4.74·17-s + 4·19-s − 20-s − 22-s − 2.74·23-s + 25-s − 4.74·26-s + 28-s − 2·29-s + 4·31-s + 32-s + 4.74·34-s − 35-s + 4.74·37-s + 4·38-s − 40-s − 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 1.31·13-s + 0.267·14-s + 0.250·16-s + 1.15·17-s + 0.917·19-s − 0.223·20-s − 0.213·22-s − 0.572·23-s + 0.200·25-s − 0.930·26-s + 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.813·34-s − 0.169·35-s + 0.780·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.951771242\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.951771242\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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
−7.75011189524773815080149054096, −7.38277780844422831690926755468, −6.54310921831661339438175862641, −5.61323166799819620248324237171, −5.11601226313032921379557658844, −4.45145099155886214723814103413, −3.58978985418995631824254397101, −2.86498404271618926306434128665, −2.01516156729491672624801330662, −0.78457194623569784087579954743,
0.78457194623569784087579954743, 2.01516156729491672624801330662, 2.86498404271618926306434128665, 3.58978985418995631824254397101, 4.45145099155886214723814103413, 5.11601226313032921379557658844, 5.61323166799819620248324237171, 6.54310921831661339438175862641, 7.38277780844422831690926755468, 7.75011189524773815080149054096