| L(s) = 1 | − 4i·2-s − 16·4-s + 148i·7-s + 64i·8-s − 384·11-s − 334i·13-s + 592·14-s + 256·16-s + 576i·17-s + 664·19-s + 1.53e3i·22-s + 3.84e3i·23-s − 1.33e3·26-s − 2.36e3i·28-s + 96·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.14i·7-s + 0.353i·8-s − 0.956·11-s − 0.548i·13-s + 0.807·14-s + 0.250·16-s + 0.483i·17-s + 0.421·19-s + 0.676i·22-s + 1.51i·23-s − 0.387·26-s − 0.570i·28-s + 0.0211·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6150953627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6150953627\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 148iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 384T + 1.61e5T^{2} \) |
| 13 | \( 1 + 334iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 576iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 664T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.84e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 96T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.79e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 6.72e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.48e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.92e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 7.77e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.66e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.68e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.80e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.64e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 8.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.99e4iT - 8.58e9T^{2} \) |
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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
−9.965096712655577045952845939012, −9.105032260247402833706613158926, −8.270360642233371814865417229996, −7.31737471736761941968561493578, −5.66568585321206979654863011203, −5.31571790302174796653268452805, −3.75842895960147398168167904966, −2.73533929142758571165712593847, −1.74998504519263957003663357000, −0.16454095383701409999386283611,
1.02641140103927649285006683191, 2.76456391701795213468103585635, 4.14503645007329224063603529960, 4.91483544490554812025275985515, 6.12051843623748743059620516990, 7.11077569551163690015173916383, 7.72498097277686902083697945734, 8.722315284865624994013131989467, 9.778194363185077727071770066547, 10.52996242230437359961534617030
