| L(s) = 1 | + (−0.434 + 0.565i)2-s + (1.99 + 2.23i)3-s + (0.903 + 3.37i)4-s + (−4.79 − 4.20i)5-s + (−2.13 + 0.156i)6-s + (−7.75 + 6.79i)7-s + (−4.93 − 2.04i)8-s + (−1.03 + 8.94i)9-s + (4.46 − 0.887i)10-s + (0.196 − 2.99i)11-s + (−5.75 + 8.75i)12-s + (2.50 − 0.671i)13-s + (−0.480 − 7.33i)14-s + (−0.152 − 19.1i)15-s + (−8.79 + 5.07i)16-s + (15.4 + 7.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.217 + 0.282i)2-s + (0.665 + 0.746i)3-s + (0.225 + 0.843i)4-s + (−0.959 − 0.841i)5-s + (−0.355 + 0.0261i)6-s + (−1.10 + 0.971i)7-s + (−0.616 − 0.255i)8-s + (−0.114 + 0.993i)9-s + (0.446 − 0.0887i)10-s + (0.0178 − 0.272i)11-s + (−0.479 + 0.729i)12-s + (0.192 − 0.0516i)13-s + (−0.0343 − 0.524i)14-s + (−0.0101 − 1.27i)15-s + (−0.549 + 0.317i)16-s + (0.906 + 0.423i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.205124 + 0.983808i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.205124 + 0.983808i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.99 - 2.23i)T \) |
| 17 | \( 1 + (-15.4 - 7.19i)T \) |
| good | 2 | \( 1 + (0.434 - 0.565i)T + (-1.03 - 3.86i)T^{2} \) |
| 5 | \( 1 + (4.79 + 4.20i)T + (3.26 + 24.7i)T^{2} \) |
| 7 | \( 1 + (7.75 - 6.79i)T + (6.39 - 48.5i)T^{2} \) |
| 11 | \( 1 + (-0.196 + 2.99i)T + (-119. - 15.7i)T^{2} \) |
| 13 | \( 1 + (-2.50 + 0.671i)T + (146. - 84.5i)T^{2} \) |
| 19 | \( 1 + (1.04 + 2.51i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-5.65 - 11.4i)T + (-322. + 419. i)T^{2} \) |
| 29 | \( 1 + (-10.1 - 30.0i)T + (-667. + 511. i)T^{2} \) |
| 31 | \( 1 + (-3.41 - 52.0i)T + (-952. + 125. i)T^{2} \) |
| 37 | \( 1 + (13.1 + 19.7i)T + (-523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (21.8 + 7.43i)T + (1.33e3 + 1.02e3i)T^{2} \) |
| 43 | \( 1 + (-75.9 + 9.99i)T + (1.78e3 - 478. i)T^{2} \) |
| 47 | \( 1 + (7.83 - 29.2i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (29.0 + 70.0i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-65.2 + 50.0i)T + (900. - 3.36e3i)T^{2} \) |
| 61 | \( 1 + (-43.9 - 50.1i)T + (-485. + 3.68e3i)T^{2} \) |
| 67 | \( 1 + (-101. - 58.6i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-8.17 - 12.2i)T + (-1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (11.5 - 58.1i)T + (-4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (7.05 - 107. i)T + (-6.18e3 - 814. i)T^{2} \) |
| 83 | \( 1 + (117. + 90.0i)T + (1.78e3 + 6.65e3i)T^{2} \) |
| 89 | \( 1 + (-2.00 + 2.00i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (44.6 + 131. i)T + (-7.46e3 + 5.72e3i)T^{2} \) |
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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
−12.77248076662307098212084423793, −12.46256765164354669749971872769, −11.27712065429242123029895539987, −9.790755737599356795240617831585, −8.688124485274242227639613577632, −8.423702113350345588624254712058, −7.08505213047324947815554600930, −5.42953023846719337323519464705, −3.84163502455684529947555628850, −3.01284647699897709689492426313,
0.62871567515933196323902974505, 2.70854666706157468232942352953, 3.86375441808366191244334010898, 6.20375014018738724616200394376, 7.02243175721720705319508280638, 7.87830675366955666043900444477, 9.451035630127573967026352666551, 10.20227274033898128971456290336, 11.28920687669057934012472022093, 12.21749828131782553337471115713
