| L(s) = 1 | + 2.82i·2-s + (8.46 + 3.06i)3-s − 8.00·4-s + 37.9i·5-s + (−8.66 + 23.9i)6-s + 17.2·7-s − 22.6i·8-s + (62.2 + 51.8i)9-s − 107.·10-s + 113. i·11-s + (−67.7 − 24.5i)12-s + 95.5·13-s + 48.8i·14-s + (−116. + 321. i)15-s + 64.0·16-s − 276. i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.940 + 0.340i)3-s − 0.500·4-s + 1.51i·5-s + (−0.240 + 0.664i)6-s + 0.352·7-s − 0.353i·8-s + (0.768 + 0.640i)9-s − 1.07·10-s + 0.935i·11-s + (−0.470 − 0.170i)12-s + 0.565·13-s + 0.249i·14-s + (−0.516 + 1.42i)15-s + 0.250·16-s − 0.955i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.373010522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.373010522\) |
| \(L(3)\) |
|
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 | \( 1 + (-8.46 - 3.06i)T \) |
| 29 | \( 1 - 156. iT \) |
| good | 5 | \( 1 - 37.9iT - 625T^{2} \) |
| 7 | \( 1 - 17.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 113. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 95.5T + 2.85e4T^{2} \) |
| 17 | \( 1 + 276. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 386.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 42.3iT - 2.79e5T^{2} \) |
| 31 | \( 1 - 204.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 339.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 169. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 682.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.17e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.89e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.66e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.28e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.47e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.87e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.35e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 9.88e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.81e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.13e4T + 8.85e7T^{2} \) |
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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
−12.78976198893656457194020245136, −11.23227348473356362258662444544, −10.30787141954063704844246761193, −9.431201390769028996926431755498, −8.219850119279884535938705447789, −7.29626706779495994885408941991, −6.50156850264714919941854698577, −4.77796703790819554002717174153, −3.52750287917302128694455108241, −2.22692672953386390406925440388,
0.822539040700188355301660572787, 1.90467886780767714216923098892, 3.60409070065793987150413423083, 4.64738703459551265492454848318, 6.13832912900250440022176437476, 8.122288054578454694192179625692, 8.504572388797337617084639784716, 9.329951313620342606971768093923, 10.60269075701372973407379202412, 11.80093416599074787043535723445
