L(s) = 1 | + (−0.777 + 0.974i)2-s + (−0.623 − 0.781i)3-s + (0.0990 + 0.433i)4-s + (1.72 + 0.829i)5-s + 1.24·6-s + 1.35·7-s + (−2.74 − 1.32i)8-s + (−0.222 + 0.974i)9-s + (−2.14 + 1.03i)10-s + (−1.30 + 5.70i)11-s + (0.277 − 0.347i)12-s + (5.35 + 2.57i)13-s + (−1.05 + 1.32i)14-s + (−0.425 − 1.86i)15-s + (2.62 − 1.26i)16-s + (−2.24 + 1.08i)17-s + ⋯ |
L(s) = 1 | + (−0.549 + 0.689i)2-s + (−0.359 − 0.451i)3-s + (0.0495 + 0.216i)4-s + (0.770 + 0.370i)5-s + 0.509·6-s + 0.512·7-s + (−0.971 − 0.467i)8-s + (−0.0741 + 0.324i)9-s + (−0.679 + 0.327i)10-s + (−0.392 + 1.71i)11-s + (0.0801 − 0.100i)12-s + (1.48 + 0.714i)13-s + (−0.281 + 0.353i)14-s + (−0.109 − 0.481i)15-s + (0.655 − 0.315i)16-s + (−0.544 + 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.661440 + 0.534139i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.661440 + 0.534139i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.623 + 0.781i)T \) |
| 43 | \( 1 + (6.24 - 1.99i)T \) |
good | 2 | \( 1 + (0.777 - 0.974i)T + (-0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-1.72 - 0.829i)T + (3.11 + 3.90i)T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + (1.30 - 5.70i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-5.35 - 2.57i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (2.24 - 1.08i)T + (10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 + (1.23 + 5.41i)T + (-17.1 + 8.24i)T^{2} \) |
| 23 | \( 1 + (-1.88 + 8.24i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-2.10 + 2.64i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (1.77 - 2.22i)T + (-6.89 - 30.2i)T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 41 | \( 1 + (0.538 - 0.674i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (0.228 + 1.00i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-10.2 + 4.92i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (3.65 - 1.76i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (2.71 + 3.40i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + (2.43 + 10.6i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (0.0658 + 0.288i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-6.35 - 3.06i)T + (45.5 + 57.0i)T^{2} \) |
| 79 | \( 1 + 4.55T + 79T^{2} \) |
| 83 | \( 1 + (3.43 + 4.30i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-2.45 - 3.07i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + (3.41 - 14.9i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−13.44317631393696563056686010620, −12.65146254314257118109603982929, −11.47714854934710650360732874748, −10.38692609013698572332218989278, −9.124141193860589424364003121330, −8.149406182198284233513954425860, −6.84168964788404753882044442594, −6.38120679614952777854097342877, −4.60287756590835366947682183898, −2.24236084514485675507767684105,
1.32004901995573857428150453640, 3.37195381252403208855384968488, 5.55771703745341913511390894906, 5.91827895703635024856037560151, 8.275604218764341849474530600048, 9.053927277322688287352832217653, 10.17346217835495973895425552603, 11.01616210325992882919996353933, 11.53846729260139921450778786506, 13.15735966480111001676827743743