L(s) = 1 | + (28.8 + 49.9i)5-s + (−78.4 + 135. i)7-s + (−109. + 188. i)11-s + (293. + 508. i)13-s − 2.12e3·17-s + 29.7·19-s + (1.58e3 + 2.74e3i)23-s + (−97.9 + 169. i)25-s + (1.82e3 − 3.15e3i)29-s + (−3.16e3 − 5.48e3i)31-s − 9.04e3·35-s − 8.43e3·37-s + (5.07e3 + 8.78e3i)41-s + (3.42e3 − 5.93e3i)43-s + (1.22e4 − 2.11e4i)47-s + ⋯ |
L(s) = 1 | + (0.515 + 0.892i)5-s + (−0.605 + 1.04i)7-s + (−0.271 + 0.470i)11-s + (0.482 + 0.835i)13-s − 1.78·17-s + 0.0189·19-s + (0.625 + 1.08i)23-s + (−0.0313 + 0.0543i)25-s + (0.402 − 0.696i)29-s + (−0.591 − 1.02i)31-s − 1.24·35-s − 1.01·37-s + (0.471 + 0.816i)41-s + (0.282 − 0.489i)43-s + (0.807 − 1.39i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8253318359\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8253318359\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-28.8 - 49.9i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (78.4 - 135. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (109. - 188. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-293. - 508. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 2.12e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 29.7T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.58e3 - 2.74e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.82e3 + 3.15e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (3.16e3 + 5.48e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 8.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-5.07e3 - 8.78e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-3.42e3 + 5.93e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.22e4 + 2.11e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 2.54e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (1.03e4 + 1.79e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.27e4 - 3.94e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (9.54e3 + 1.65e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-1.70e4 + 2.94e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (5.42e4 - 9.39e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 1.10e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.44e4 + 2.49e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−11.27592953852710107565448934764, −10.37053223646739159736351980950, −9.343516944916308752721478066628, −8.772264515626891262506139387931, −7.21974949389650567932196666945, −6.46340645157825517047273952067, −5.61751901594843791104395743963, −4.18033092384279651113195282633, −2.75278378713196799830608151815, −1.98591317079852571484875643988,
0.21495552950384213996765839803, 1.24768920583024728301539743938, 2.89349900983799240890370084098, 4.19037494896342117939083252947, 5.21776316017294781391161763792, 6.35894533435485944737733922806, 7.28674641483902730173229552683, 8.660100642816801662717676773159, 9.098067649476005483627147303885, 10.61839525813005262602810900740