| L(s) = 1 | + (−1.34 − 0.776i)5-s + (6.19 + 10.7i)7-s + (−12.7 + 7.34i)11-s + (5.40 − 9.35i)13-s + 28.9i·17-s − 3.60·19-s + (−12.7 − 7.34i)23-s + (−11.2 − 19.5i)25-s + (−24.3 + 14.0i)29-s + (4 − 6.92i)31-s − 19.2i·35-s + 22.5·37-s + (21.7 + 12.5i)41-s + (−26.5 − 46.0i)43-s + (−14.6 + 8.48i)47-s + ⋯ |
| L(s) = 1 | + (−0.268 − 0.155i)5-s + (0.885 + 1.53i)7-s + (−1.15 + 0.668i)11-s + (0.415 − 0.719i)13-s + 1.70i·17-s − 0.189·19-s + (−0.553 − 0.319i)23-s + (−0.451 − 0.782i)25-s + (−0.840 + 0.485i)29-s + (0.129 − 0.223i)31-s − 0.549i·35-s + 0.609·37-s + (0.531 + 0.306i)41-s + (−0.618 − 1.07i)43-s + (−0.312 + 0.180i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6664754092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6664754092\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.34 + 0.776i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-6.19 - 10.7i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (12.7 - 7.34i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-5.40 + 9.35i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 28.9iT - 289T^{2} \) |
| 19 | \( 1 + 3.60T + 361T^{2} \) |
| 23 | \( 1 + (12.7 + 7.34i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (24.3 - 14.0i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 22.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-21.7 - 12.5i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (26.5 + 46.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (14.6 - 8.48i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 84.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (78.8 + 45.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (20.5 - 35.6i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 16.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 71.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + (23.3 + 40.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-13.2 + 7.65i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 78.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (45.5 + 78.9i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.911321338293167730963026526687, −8.834466113033688301281582625609, −8.082185474618395689059259182621, −7.906485260273567135369994279216, −6.35433134743195948725960323696, −5.61141933411234810252635804525, −4.93906649013195526547578016642, −3.85934055445010938370489542871, −2.51856184171464418739147685321, −1.76919610286636593790306184069,
0.18688006939532403264718423005, 1.42237559260859172327735412293, 2.84902591846775453510515097215, 3.94304725018647792081863143377, 4.68938874091601127525829534135, 5.61620228614255672657812451240, 6.78745539784246027027884794580, 7.73251782943800794263657740012, 7.82032226787486234212498866701, 9.125641475147384120396651341916
