L(s) = 1 | − 8.42·2-s + 38.9·4-s + 30.3·5-s − 197.·7-s − 58.7·8-s − 255.·10-s + 604.·11-s − 893.·13-s + 1.66e3·14-s − 752.·16-s − 1.08e3·17-s + 2.02e3·19-s + 1.18e3·20-s − 5.09e3·22-s − 1.32e3·23-s − 2.20e3·25-s + 7.52e3·26-s − 7.70e3·28-s − 2.51e3·29-s + 8.78e3·31-s + 8.21e3·32-s + 9.16e3·34-s − 6.00e3·35-s − 1.30e4·37-s − 1.70e4·38-s − 1.78e3·40-s − 1.06e4·41-s + ⋯ |
L(s) = 1 | − 1.48·2-s + 1.21·4-s + 0.543·5-s − 1.52·7-s − 0.324·8-s − 0.809·10-s + 1.50·11-s − 1.46·13-s + 2.27·14-s − 0.734·16-s − 0.913·17-s + 1.28·19-s + 0.661·20-s − 2.24·22-s − 0.521·23-s − 0.704·25-s + 2.18·26-s − 1.85·28-s − 0.556·29-s + 1.64·31-s + 1.41·32-s + 1.35·34-s − 0.828·35-s − 1.56·37-s − 1.91·38-s − 0.176·40-s − 0.989·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5745272933\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5745272933\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + 3.48e3T \) |
good | 2 | \( 1 + 8.42T + 32T^{2} \) |
| 5 | \( 1 - 30.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 197.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 604.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 893.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.02e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.30e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.06e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.39e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.74e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 1.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.82e5T + 8.58e9T^{2} \) |
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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
−9.716798821108626508164914691823, −9.441634635370389157866676629423, −8.615359921164346582238983592644, −7.23340486869577277861561562352, −6.82908798681387741652965585490, −5.80773600754266134773688080740, −4.23294615704009974847570544269, −2.85327657969459382920100441672, −1.73028449356616316053218843007, −0.46818198276238427376349439145,
0.46818198276238427376349439145, 1.73028449356616316053218843007, 2.85327657969459382920100441672, 4.23294615704009974847570544269, 5.80773600754266134773688080740, 6.82908798681387741652965585490, 7.23340486869577277861561562352, 8.615359921164346582238983592644, 9.441634635370389157866676629423, 9.716798821108626508164914691823