| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 7.93i·5-s + 0.453·7-s + 2.82i·8-s + 11.2·10-s − 14.1i·11-s − 12.8·13-s − 0.641i·14-s + 4.00·16-s + 32.9i·17-s − 0.405·19-s − 15.8i·20-s − 19.9·22-s − 14.4i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.58i·5-s + 0.0648·7-s + 0.353i·8-s + 1.12·10-s − 1.28i·11-s − 0.984·13-s − 0.0458i·14-s + 0.250·16-s + 1.93i·17-s − 0.0213·19-s − 0.793i·20-s − 0.906·22-s − 0.629i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.006183335234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006183335234\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 7.93iT - 25T^{2} \) |
| 7 | \( 1 - 0.453T + 49T^{2} \) |
| 11 | \( 1 + 14.1iT - 121T^{2} \) |
| 13 | \( 1 + 12.8T + 169T^{2} \) |
| 17 | \( 1 - 32.9iT - 289T^{2} \) |
| 19 | \( 1 + 0.405T + 361T^{2} \) |
| 23 | \( 1 + 14.4iT - 529T^{2} \) |
| 29 | \( 1 - 26.2iT - 841T^{2} \) |
| 31 | \( 1 - 24.9T + 961T^{2} \) |
| 37 | \( 1 + 7.68T + 1.36e3T^{2} \) |
| 41 | \( 1 + 24.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 17.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 47.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 0.261iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 55.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 104.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 64.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 109. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 62.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 19.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 57.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 103. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 56.2T + 9.40e3T^{2} \) |
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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
−10.13521854203673767999711243284, −8.952584309401744611188088538099, −8.220539540347541771949969042671, −7.32657481064012580910428770213, −6.38475128426038088831270835640, −5.76817829549146263386534900241, −4.44095854577452358297321344436, −3.40822052990353947101598036185, −2.85257732467396786020363462281, −1.74007399070932027535410662092,
0.00177187133409916509943273341, 1.24823129668362277160322138544, 2.62230333367050921800736423188, 4.30899610481086196630257721176, 4.82599089698423367029838233124, 5.33208883630797156036638100533, 6.54322023988547244027904873800, 7.53902427665093741688879802029, 7.888627206351551472182985668332, 9.069616487539839316925166748654
