L(s) = 1 | + (0.900 + 0.433i)2-s + (−0.433 − 1.90i)3-s + (0.623 + 0.781i)4-s + (0.433 − 1.90i)6-s + (−0.974 + 0.222i)7-s + (0.222 + 0.974i)8-s + (−2.52 + 1.21i)9-s + (−0.781 − 0.376i)11-s + (1.21 − 1.52i)12-s + (−1.62 − 0.781i)13-s + (−0.974 − 0.222i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 2.80·18-s + (0.846 + 1.75i)21-s + (−0.541 − 0.678i)22-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)2-s + (−0.433 − 1.90i)3-s + (0.623 + 0.781i)4-s + (0.433 − 1.90i)6-s + (−0.974 + 0.222i)7-s + (0.222 + 0.974i)8-s + (−2.52 + 1.21i)9-s + (−0.781 − 0.376i)11-s + (1.21 − 1.52i)12-s + (−1.62 − 0.781i)13-s + (−0.974 − 0.222i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 2.80·18-s + (0.846 + 1.75i)21-s + (−0.541 − 0.678i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2673660014\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2673660014\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 7 | \( 1 + (0.974 - 0.222i)T \) |
| 17 | \( 1 + (-0.623 + 0.781i)T \) |
good | 3 | \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \) |
| 5 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 11 | \( 1 + (0.781 + 0.376i)T + (0.623 + 0.781i)T^{2} \) |
| 13 | \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.541 - 0.678i)T + (-0.222 + 0.974i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + 1.94T + T^{2} \) |
| 37 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 53 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 61 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.21 + 1.52i)T + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 + 0.867T + T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−7.80699470881939644249981268743, −7.38393170057695649668312455791, −7.08531067715683378001581808376, −6.01689171495149065354658698628, −5.54395945562230092680469085991, −5.07336401313689751857624199112, −3.27409030342682316919110011505, −2.80238158492081189414608585623, −1.90841375903739943625478683330, −0.10767528396247352350189559937,
2.34767431397432780283828937594, 3.13828060962826720155907845116, 3.96591024590629382504147936550, 4.48001538875226293895249479961, 5.28461874352438975861129440212, 5.78747084039527516690772350047, 6.68754954247841581002982176960, 7.57032090135059956705884929474, 8.962620039866151314514932569728, 9.571211632777080821384690869910