| L(s) = 1 | − 4i·2-s − 16·4-s + 25·5-s + (−124. + 34.7i)7-s + 64i·8-s − 100i·10-s − 250. i·11-s + 257. i·13-s + (139. + 499. i)14-s + 256·16-s − 1.65e3·17-s + 117. i·19-s − 400·20-s − 1.00e3·22-s − 3.22e3i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447·5-s + (−0.963 + 0.268i)7-s + 0.353i·8-s − 0.316i·10-s − 0.625i·11-s + 0.421i·13-s + (0.189 + 0.681i)14-s + 0.250·16-s − 1.38·17-s + 0.0748i·19-s − 0.223·20-s − 0.442·22-s − 1.27i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.367254005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367254005\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 25T \) |
| 7 | \( 1 + (124. - 34.7i)T \) |
| good | 11 | \( 1 + 250. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 257. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.65e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 117. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.22e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.17e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.32e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 8.37e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.46e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.33e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 9.24e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.96e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.64e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.68e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.40e4iT - 8.58e9T^{2} \) |
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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
−9.886918674471704861385455099489, −8.941463385584470584532865873618, −8.479196659409785845044774892895, −6.85447315170971087283826960903, −6.25322849744826378424136095639, −5.09775822796191360147416774042, −4.01120916519832113066006529962, −2.92455960859360907825514124917, −2.07633582839895236067100154676, −0.64809336888673072532399766695,
0.45699149695329614440894755525, 2.02673929104209399246744308882, 3.34764216249412887105583332035, 4.42878530506830058370616263481, 5.47935412498142404858373939679, 6.40952565023545555507335236067, 7.04799819349890571031381609182, 7.999017251947045488650160390852, 9.132649324009379438817488194624, 9.666080009552955243084237801340
