L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)16-s + (0.5 + 1.53i)17-s + (1.30 − 0.951i)19-s + (0.309 + 0.951i)22-s + (−0.809 + 0.587i)25-s − 32-s − 1.61·34-s + (0.499 + 1.53i)38-s + (0.5 − 0.363i)41-s + 0.618·43-s − 0.999·44-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)16-s + (0.5 + 1.53i)17-s + (1.30 − 0.951i)19-s + (0.309 + 0.951i)22-s + (−0.809 + 0.587i)25-s − 32-s − 1.61·34-s + (0.499 + 1.53i)38-s + (0.5 − 0.363i)41-s + 0.618·43-s − 0.999·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8273706295\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8273706295\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-0.809 + 0.587i)T \) |
good | 5 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + 0.618T + T^{2} \) |
| 97 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.46756545423740723141866820341, −9.468747824644091638777667486901, −8.943948421324790125508300250963, −7.954801449549138117505299441707, −7.26701973604653971690362006205, −6.19026843786343710257042285668, −5.62774743081545042905803644765, −4.41564810886384052064988578427, −3.40497491778558501238876564787, −1.36416317590207705578583067242,
1.31857430434776127440643131250, 2.69174794412313188263532661467, 3.75162344067135490516729953060, 4.72520165683107944567022386793, 5.78191450161745878799906852727, 7.22384830096375345010998812755, 7.81564347402769255457210750099, 8.981819456720351159032683222625, 9.650349743144352185016882181236, 10.17016646836883994171040440533