L(s) = 1 | + (−1.12 + 0.817i)2-s + (−0.309 − 0.951i)3-s + (−0.0207 + 0.0638i)4-s + (1.12 + 0.817i)6-s + (−0.394 + 1.21i)7-s + (−0.888 − 2.73i)8-s + (−0.809 + 0.587i)9-s + (−1.20 − 3.09i)11-s + 0.0671·12-s + (−1.14 + 0.833i)13-s + (−0.548 − 1.68i)14-s + (3.12 + 2.26i)16-s + (4.04 + 2.93i)17-s + (0.429 − 1.32i)18-s + (0.0488 + 0.150i)19-s + ⋯ |
L(s) = 1 | + (−0.795 + 0.577i)2-s + (−0.178 − 0.549i)3-s + (−0.0103 + 0.0319i)4-s + (0.459 + 0.333i)6-s + (−0.149 + 0.459i)7-s + (−0.313 − 0.966i)8-s + (−0.269 + 0.195i)9-s + (−0.362 − 0.931i)11-s + 0.0193·12-s + (−0.318 + 0.231i)13-s + (−0.146 − 0.451i)14-s + (0.780 + 0.567i)16-s + (0.981 + 0.712i)17-s + (0.101 − 0.311i)18-s + (0.0112 + 0.0345i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.765654 + 0.263039i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.765654 + 0.263039i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (1.20 + 3.09i)T \) |
good | 2 | \( 1 + (1.12 - 0.817i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (0.394 - 1.21i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (1.14 - 0.833i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.04 - 2.93i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.0488 - 0.150i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 + (-1.93 + 5.96i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.46 - 1.79i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.45 + 4.46i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.34 - 7.21i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 + (-2.54 - 7.82i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.57 + 5.50i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.50 + 7.70i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-11.5 - 8.40i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 + (-5.48 - 3.98i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.67 - 8.23i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.05 + 1.49i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.18 - 5.94i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + (-5.18 + 3.76i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.08706660176464869373538247876, −9.279427683256549803969859129154, −8.411037261811816203679822330056, −7.912673186049014341910434368970, −6.99548349617447977758840391373, −6.16440144570047287366593540252, −5.35686636213358670313945184102, −3.79728187860782538046485939383, −2.67208207207784548339927029294, −0.880362008215214541359063878965,
0.793831081968000064909716372896, 2.33679538605375073096955659830, 3.50720843450651045119153615217, 4.93239756559607382269083861504, 5.40557095580294393834597968615, 6.88437807563762315816063484724, 7.69245886792352146801712451956, 8.799857814291111002902251245869, 9.468535618896668578091232827704, 10.28915360944200148121241083056