| L(s) = 1 | + 2.82i·2-s + (8.33 − 3.38i)3-s − 8.00·4-s − 28.5i·5-s + (9.58 + 23.5i)6-s − 63.2·7-s − 22.6i·8-s + (58.0 − 56.5i)9-s + 80.7·10-s + 141. i·11-s + (−66.6 + 27.1i)12-s − 260.·13-s − 178. i·14-s + (−96.7 − 238. i)15-s + 64.0·16-s + 133. i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.926 − 0.376i)3-s − 0.500·4-s − 1.14i·5-s + (0.266 + 0.655i)6-s − 1.29·7-s − 0.353i·8-s + (0.716 − 0.697i)9-s + 0.807·10-s + 1.16i·11-s + (−0.463 + 0.188i)12-s − 1.54·13-s − 0.912i·14-s + (−0.430 − 1.05i)15-s + 0.250·16-s + 0.460i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.376i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2447580833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2447580833\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 + (-8.33 + 3.38i)T \) |
| 29 | \( 1 - 156. iT \) |
| good | 5 | \( 1 + 28.5iT - 625T^{2} \) |
| 7 | \( 1 + 63.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 141. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 260.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 133. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 555.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 511. iT - 2.79e5T^{2} \) |
| 31 | \( 1 + 1.22e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.63e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.38e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.01e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.13e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.54e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.10e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.00e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 897.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 7.65e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.71e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.18e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 9.78e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.22e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 4.45e3T + 8.85e7T^{2} \) |
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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
−12.28808746615106013673848613769, −9.931891969849976760712636490852, −9.473022380006135213644749895413, −8.510590902107784032195525253994, −7.41525044461274745009956195539, −6.57722644584348018000781619178, −5.00925009424557996177295719661, −3.83675582990279941166378180304, −2.06640745247344271286909653315, −0.07428339248339567560315056162,
2.58321382841077735420907780898, 3.04455993930256599633472694973, 4.35863115577941900736338143605, 6.23899146845667618998045762959, 7.37544181800983446156064465487, 8.693154335329688471553022471695, 9.696297370849731811022440198797, 10.34488671528648356064588985302, 11.23649591684215732550903053172, 12.68881947691747015420057349178
