| 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
| \(L(3)\) |
\(\approx\) |
\(0.4179892640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4179892640\) |
| \(L(\frac{7}{2})\) |
|
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
−10.29283395367664719238409837565, −9.152947816719079385095798070316, −8.689308735128490496094994631313, −7.64432221046987636845329623009, −6.26270249652503230200872854519, −5.48005775878713493461424416840, −4.20903552733793569109745170373, −3.10136034347148308274188669544, −1.52043224272315096071483687174, −0.11578460892880684519408866103,
1.33846540846759387419794371497, 3.31470154108191154795789727307, 3.65363179086979786611077689924, 5.31716310517357319962077182326, 6.54232470549436763564857343192, 7.39510940684199081370413153002, 8.046781854375936424044142645991, 9.667017060548522486381309892856, 10.23381344611884496491259223236, 11.07799293451589192719092608784
