| L(s) = 1 | + (2.05 + 1.18i)2-s + (3.39 + 5.88i)3-s + (−61.1 − 105. i)4-s + (−176. + 101. i)5-s + 16.1i·6-s + (721. + 1.24e3i)7-s − 593. i·8-s + (1.07e3 − 1.85e3i)9-s − 482.·10-s + 8.10e3·11-s + (415. − 720. i)12-s + (−3.44e3 + 1.99e3i)13-s + 3.42e3i·14-s + (−1.19e3 − 691. i)15-s + (−7.12e3 + 1.23e4i)16-s + (1.83e4 + 1.06e4i)17-s + ⋯ |
| L(s) = 1 | + (0.181 + 0.104i)2-s + (0.0726 + 0.125i)3-s + (−0.478 − 0.827i)4-s + (−0.630 + 0.364i)5-s + 0.0304i·6-s + (0.794 + 1.37i)7-s − 0.410i·8-s + (0.489 − 0.847i)9-s − 0.152·10-s + 1.83·11-s + (0.0694 − 0.120i)12-s + (−0.435 + 0.251i)13-s + 0.333i·14-s + (−0.0916 − 0.0529i)15-s + (−0.435 + 0.753i)16-s + (0.907 + 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.89618 + 0.315975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89618 + 0.315975i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-3.10e4 - 3.06e5i)T \) |
| good | 2 | \( 1 + (-2.05 - 1.18i)T + (64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-3.39 - 5.88i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (176. - 101. i)T + (3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-721. - 1.24e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 8.10e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (3.44e3 - 1.99e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.83e4 - 1.06e4i)T + (2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.64e4 + 2.10e4i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + 5.48e3iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.31e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 8.15e4iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (2.15e5 + 3.73e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + 7.20e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.67e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-9.45e4 + 1.63e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.06e6 + 6.15e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.75e6 + 1.01e6i)T + (1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.89e6 - 3.28e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.68e6 - 2.91e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 2.71e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-6.51e6 + 3.76e6i)T + (9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (4.14e6 - 7.17e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (6.98e6 + 4.03e6i)T + (2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 2.83e6iT - 8.07e13T^{2} \) |
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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
−14.88335104690106510158091111860, −14.21653434125672004943603425759, −12.23205860141910901636861251170, −11.50680860550636377084077963087, −9.663974207782865463300910354432, −8.814298660743020172494570113858, −6.84133875728513918846948088767, −5.36936352140827492804240949463, −3.80576650108963280917760430549, −1.31610355435017552544992909791,
1.09483023684282844640610116835, 3.76293298502271655926361578235, 4.68544270092611426331416441049, 7.39565763238506117130539301357, 8.014819232928274181943814261014, 9.732950877470659559908255169079, 11.43978483343701243195870747548, 12.28727595443024063356836612944, 13.76230809862238156401490718257, 14.32416350986742647114246147806
