| L(s) = 1 | + (−2.73 − 0.732i)2-s + (4.17 + 2.41i)3-s + (6.92 + 4i)4-s + (35.6 − 9.55i)5-s + (−9.65 − 9.65i)6-s + (−13.4 + 23.2i)7-s + (−15.9 − 16i)8-s + (−28.8 − 49.9i)9-s − 104.·10-s + 159. i·11-s + (19.3 + 33.4i)12-s + (274. − 73.6i)13-s + (53.8 − 53.8i)14-s + (172. + 46.0i)15-s + (31.9 + 55.4i)16-s + (46.7 − 174. i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.464 + 0.268i)3-s + (0.433 + 0.250i)4-s + (1.42 − 0.382i)5-s + (−0.268 − 0.268i)6-s + (−0.274 + 0.475i)7-s + (−0.249 − 0.250i)8-s + (−0.356 − 0.617i)9-s − 1.04·10-s + 1.32i·11-s + (0.134 + 0.232i)12-s + (1.62 − 0.435i)13-s + (0.274 − 0.274i)14-s + (0.764 + 0.204i)15-s + (0.124 + 0.216i)16-s + (0.161 − 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0332i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.74707 + 0.0290188i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74707 + 0.0290188i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.73 + 0.732i)T \) |
| 37 | \( 1 + (-133. - 1.36e3i)T \) |
| good | 3 | \( 1 + (-4.17 - 2.41i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-35.6 + 9.55i)T + (541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (13.4 - 23.2i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 - 159. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-274. + 73.6i)T + (2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-46.7 + 174. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-159. + 42.7i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-601. - 601. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (-92.6 + 92.6i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (149. - 149. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (1.55e3 + 899. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.04e3 + 1.04e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.88e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.90e3 + 3.30e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-1.43e3 + 5.34e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-645. - 2.40e3i)T + (-1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (4.33e3 + 2.50e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (1.84e3 - 3.19e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 - 8.11e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.84e3 + 494. i)T + (3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (2.03e3 + 3.52e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.47e4 + 3.94e3i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (1.49e3 + 1.49e3i)T + 8.85e7iT^{2} \) |
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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
−13.71396636497074547822982672915, −12.83589979185945567133519813475, −11.52066579996410037665428202680, −9.964400177481763796111904450939, −9.416055679018259663290805428265, −8.519926880799081491181857864003, −6.70519533926811603598729632107, −5.42308930261637750491711821563, −3.12892818232717664593083432551, −1.50327477803118834330886703015,
1.41604506766645132291280955823, 3.05512463595180940245394355273, 5.75809972968441576901086049196, 6.63787696703626461332068206316, 8.279538473436801290646556658957, 9.095029431946541955490828049261, 10.48059802467160167335740279921, 11.09361838783289269332736981439, 13.27118869310194046109023833309, 13.71770452187569404412813727381
