| L(s) = 1 | + (−251. − 435. i)5-s + (−776.5 + 1.34e3i)7-s + (−3.26e3 + 5.65e3i)11-s + (399.5 + 691. i)13-s + 1.15e4·17-s − 3.55e4·19-s + (5.30e4 + 9.18e4i)23-s + (−8.72e4 + 1.51e5i)25-s + (7.44e4 − 1.28e5i)29-s + (−3.89e4 − 6.74e4i)31-s + 7.80e5·35-s + 2.37e5·37-s + (1.89e5 + 3.27e5i)41-s + (1.31e4 − 2.26e4i)43-s + (−1.95e5 + 3.38e5i)47-s + ⋯ |
| L(s) = 1 | + (−0.899 − 1.55i)5-s + (−0.855 + 1.48i)7-s + (−0.740 + 1.28i)11-s + (0.0504 + 0.0873i)13-s + 0.570·17-s − 1.18·19-s + (0.908 + 1.57i)23-s + (−1.11 + 1.93i)25-s + (0.566 − 0.981i)29-s + (−0.234 − 0.406i)31-s + 3.07·35-s + 0.770·37-s + (0.428 + 0.741i)41-s + (0.0251 − 0.0435i)43-s + (−0.274 + 0.475i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3481852267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3481852267\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (251. + 435. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (776.5 - 1.34e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.26e3 - 5.65e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-399.5 - 691. i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.15e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.55e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-5.30e4 - 9.18e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-7.44e4 + 1.28e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (3.89e4 + 6.74e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 2.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.89e5 - 3.27e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.31e4 + 2.26e4i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (1.95e5 - 3.38e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.73e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (9.77e4 + 1.69e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (2.11e5 - 3.66e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (9.75e5 + 1.68e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.61e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.47e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (2.33e6 - 4.05e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (9.57e5 - 1.65e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + 2.61e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-4.15e6 + 7.18e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−9.751677371808702598089848399541, −9.254173909918936923007696421202, −8.308007382279043564789022292905, −7.53712997375137141408315726639, −6.04926101825592347874181318910, −5.11615643271927891425894801128, −4.28631117977745335131128831712, −2.87828819483965992367872968576, −1.60363516850377203127932130757, −0.11861849873382609079222431052,
0.65934733597578388365063730897, 2.86283277470631522982219118922, 3.36137315869737427421137870173, 4.39343392968123152785634240856, 6.19015001761578931595616051263, 6.86231814939535404653848003121, 7.63232084451204975705639071966, 8.595944780525810507033512023909, 10.25091217867355526383439747646, 10.65716771965395950093918305744
