L(s) = 1 | + (−1 + 1.73i)2-s + (1.63 + 2.83i)3-s + (−1.99 − 3.46i)4-s + (−9.76 − 16.9i)5-s − 6.55·6-s + (−7.16 − 12.4i)7-s + 7.99·8-s + (8.12 − 14.0i)9-s + 39.0·10-s − 40.4·11-s + (6.55 − 11.3i)12-s + (29.3 + 50.8i)13-s + 28.6·14-s + (32.0 − 55.4i)15-s + (−8 + 13.8i)16-s + (43.7 − 75.7i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.315 + 0.546i)3-s + (−0.249 − 0.433i)4-s + (−0.873 − 1.51i)5-s − 0.445·6-s + (−0.387 − 0.670i)7-s + 0.353·8-s + (0.301 − 0.521i)9-s + 1.23·10-s − 1.10·11-s + (0.157 − 0.273i)12-s + (0.626 + 1.08i)13-s + 0.547·14-s + (0.550 − 0.954i)15-s + (−0.125 + 0.216i)16-s + (0.623 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.623696 - 0.482464i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.623696 - 0.482464i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 37 | \( 1 + (-24.0 - 223. i)T \) |
good | 3 | \( 1 + (-1.63 - 2.83i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (9.76 + 16.9i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (7.16 + 12.4i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + 40.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-29.3 - 50.8i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-43.7 + 75.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (55.4 + 96.1i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 121.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (120. + 208. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 - 222.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 242.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-255. + 441. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-138. + 239. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-185. - 321. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-55.8 - 96.7i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-304. - 526. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 64.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-444. - 769. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-643. + 1.11e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (201. - 349. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−13.88085546188074493393247491680, −12.90661753758385585420009836299, −11.68115252038965650482419688675, −10.10773098882946321612775163440, −9.087868742282336266149052534378, −8.231727605184260814100920468814, −6.92659877086560009307799482759, −4.97710166721376665152160135037, −3.97987066633208370631233787498, −0.54919644440612391123735505458,
2.40452090431143133372207346355, 3.58192854373335035608437158625, 6.08843298229702989755442773077, 7.75535024864781510885634392034, 8.146260579594997999749710444152, 10.38527503243495328043913607977, 10.62242463050721607594695005900, 12.18705685217715712232531622234, 12.94162557038199320944486073926, 14.26664437741719602921346080222