L(s) = 1 | − 1.79·2-s + 1.20·4-s + 3·5-s + 1.41·8-s − 5.37·10-s + 11-s − 13-s − 4.95·16-s + 7.58·17-s + 6.58·19-s + 3.62·20-s − 1.79·22-s + 5.58·23-s + 4·25-s + 1.79·26-s + 8.16·29-s − 3.58·31-s + 6.04·32-s − 13.5·34-s + 37-s − 11.7·38-s + 4.25·40-s − 11.1·41-s + 1.58·43-s + 1.20·44-s − 10·46-s + 1.41·47-s + ⋯ |
L(s) = 1 | − 1.26·2-s + 0.604·4-s + 1.34·5-s + 0.501·8-s − 1.69·10-s + 0.301·11-s − 0.277·13-s − 1.23·16-s + 1.83·17-s + 1.51·19-s + 0.810·20-s − 0.381·22-s + 1.16·23-s + 0.800·25-s + 0.351·26-s + 1.51·29-s − 0.643·31-s + 1.06·32-s − 2.32·34-s + 0.164·37-s − 1.91·38-s + 0.672·40-s − 1.74·41-s + 0.241·43-s + 0.182·44-s − 1.47·46-s + 0.206·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.569415164\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.569415164\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 7.58T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 9.16T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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
−8.478249682641496975093706281935, −7.54744854258677867373333520729, −7.12614621071580458266347643477, −6.20690863194529074716862246155, −5.37073059993786927539341671792, −4.88678931179284937809491967345, −3.47235489106323898093215025445, −2.61349040716235079611381681774, −1.48015844081657071450979856897, −0.964284910030881116347584445651,
0.964284910030881116347584445651, 1.48015844081657071450979856897, 2.61349040716235079611381681774, 3.47235489106323898093215025445, 4.88678931179284937809491967345, 5.37073059993786927539341671792, 6.20690863194529074716862246155, 7.12614621071580458266347643477, 7.54744854258677867373333520729, 8.478249682641496975093706281935