| L(s) = 1 | + 3i·3-s + (−11.1 − 1.29i)5-s − 16.2i·7-s − 9·9-s + 40.2·11-s + 19.7i·13-s + (3.89 − 33.3i)15-s + 83.0i·17-s − 48.8·19-s + 48.6·21-s − 1.61i·23-s + (121. + 28.8i)25-s − 27i·27-s − 24.5·29-s − 12.4·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.993 − 0.116i)5-s − 0.875i·7-s − 0.333·9-s + 1.10·11-s + 0.422i·13-s + (0.0670 − 0.573i)15-s + 1.18i·17-s − 0.589·19-s + 0.505·21-s − 0.0146i·23-s + (0.973 + 0.230i)25-s − 0.192i·27-s − 0.157·29-s − 0.0719·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8089224141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8089224141\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 + (11.1 + 1.29i)T \) |
| good | 7 | \( 1 + 16.2iT - 343T^{2} \) |
| 11 | \( 1 - 40.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 83.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.61iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 24.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 325. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 367. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 204. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 61.5iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 477.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 558. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 558.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 96.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3iT - 9.12e5T^{2} \) |
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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
−9.334576536883437507354268874477, −8.648804574137999009794052815744, −7.78882206703832679856205218107, −6.90886217811255426252535055079, −6.06399141612642812599283908990, −4.64248648207209716688024475750, −4.04583543285156486142106134737, −3.41270743658268490043613980712, −1.62197455171795557844506227558, −0.23800668502428714735360319190,
1.06807196190786296545361852104, 2.48231468423548000275155317984, 3.44710796704230525234472844554, 4.56723373759716685688835866666, 5.60930422001475050703226192734, 6.65265815063035451956044789354, 7.24816523243480466512828221285, 8.297171448060028054357749731253, 8.800003285703034623056219288687, 9.740781709580575918753460013028
