| L(s) = 1 | + (1.35 + 3.76i)2-s + (−12.3 + 10.2i)4-s − 2.66·5-s − 88.8i·7-s + (−55.2 − 32.3i)8-s + (−3.61 − 10.0i)10-s − 72.9i·11-s − 173.·13-s + (334. − 120. i)14-s + (46.7 − 251. i)16-s + 423.·17-s + 153. i·19-s + (32.7 − 27.2i)20-s + (274. − 99.1i)22-s − 723. i·23-s + ⋯ |
| L(s) = 1 | + (0.339 + 0.940i)2-s + (−0.768 + 0.639i)4-s − 0.106·5-s − 1.81i·7-s + (−0.862 − 0.505i)8-s + (−0.0361 − 0.100i)10-s − 0.602i·11-s − 1.02·13-s + (1.70 − 0.616i)14-s + (0.182 − 0.983i)16-s + 1.46·17-s + 0.424i·19-s + (0.0818 − 0.0680i)20-s + (0.567 − 0.204i)22-s − 1.36i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.09291 - 0.512630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09291 - 0.512630i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 - 3.76i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.66T + 625T^{2} \) |
| 7 | \( 1 + 88.8iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 72.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 173.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 423.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 153. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 723. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 366.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 642. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 535.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.14e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 615. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 330. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.62e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.46e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.64e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.67e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 6.44e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.73e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 2.08e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 9.29e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.70e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 6.97e3T + 8.85e7T^{2} \) |
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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
−13.11263281362411988477523856922, −12.07488594128780005566942840781, −10.56221245096194048737990515893, −9.591991691805989128570425405019, −7.932535150382585784512635743762, −7.39585546558460978834180699360, −6.07583574441085805779492927778, −4.60372904589352252769149174766, −3.50933875305484591864886745151, −0.48189235055705601045811559503,
1.91189709319388451590974474890, 3.15925243047587622804431985635, 4.97442344850641632836181583272, 5.83018096171137779505308721868, 7.81492413100370455282339985916, 9.264366544421614796753692205916, 9.793755759609374243337495332569, 11.35821835831494814135137049638, 12.18242058288372671041672671627, 12.66592590979405497891950915182
