| 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
−12.21749828131782553337471115713, −11.28920687669057934012472022093, −10.20227274033898128971456290336, −9.451035630127573967026352666551, −7.87830675366955666043900444477, −7.02243175721720705319508280638, −6.20375014018738724616200394376, −3.86375441808366191244334010898, −2.70854666706157468232942352953, −0.62871567515933196323902974505,
3.01284647699897709689492426313, 3.84163502455684529947555628850, 5.42953023846719337323519464705, 7.08505213047324947815554600930, 8.423702113350345588624254712058, 8.688124485274242227639613577632, 9.790755737599356795240617831585, 11.27712065429242123029895539987, 12.46256765164354669749971872769, 12.77248076662307098212084423793
