L(s) = 1 | + (0.809 − 0.587i)2-s + (0.578 − 1.78i)3-s + (0.309 − 0.951i)4-s + (−1.13 − 1.92i)5-s + (−0.578 − 1.78i)6-s − 2.72·7-s + (−0.309 − 0.951i)8-s + (−0.409 − 0.297i)9-s + (−2.05 − 0.891i)10-s + (4.97 − 3.61i)11-s + (−1.51 − 1.10i)12-s + (−3.74 − 2.72i)13-s + (−2.20 + 1.59i)14-s + (−4.08 + 0.906i)15-s + (−0.809 − 0.587i)16-s + (1.58 + 4.88i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (0.334 − 1.02i)3-s + (0.154 − 0.475i)4-s + (−0.507 − 0.861i)5-s + (−0.236 − 0.727i)6-s − 1.02·7-s + (−0.109 − 0.336i)8-s + (−0.136 − 0.0992i)9-s + (−0.648 − 0.281i)10-s + (1.49 − 1.08i)11-s + (−0.437 − 0.317i)12-s + (−1.03 − 0.755i)13-s + (−0.588 + 0.427i)14-s + (−1.05 + 0.234i)15-s + (−0.202 − 0.146i)16-s + (0.384 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0592096 + 1.79076i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0592096 + 1.79076i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (1.13 + 1.92i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.578 + 1.78i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 + (-4.97 + 3.61i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.74 + 2.72i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.58 - 4.88i)T + (-13.7 + 9.99i)T^{2} \) |
| 23 | \( 1 + (2.94 - 2.14i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.142 - 0.437i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.363 - 1.11i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.180 - 0.131i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.45 + 1.05i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 + (-3.80 + 11.7i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.997 - 3.07i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.44 - 4.68i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.15 + 4.46i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.53 - 4.71i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.68 + 11.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.18 + 5.22i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.01 + 3.11i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.91 - 15.1i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 8.55i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.38 + 7.33i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.616840542808096126733817708765, −8.680821110512581513753815107742, −7.979793410351021405983652174781, −6.93721203853377830753696128729, −6.22519577101544365380891516823, −5.26187306460431560648326458577, −3.93086290191112225396837998084, −3.28414873560119372149066377930, −1.83240608224481032002686823209, −0.66654072770164721170003167143,
2.47574200903116624182144909613, 3.54004881498018791831238923286, 4.13456303841547227389223312399, 4.93081956671682761805388209754, 6.55583581720937163251409327575, 6.77706959383693763395021137949, 7.70583965669380019196676041818, 9.084710721645884328342819393496, 9.729969064738648744166470324664, 10.08684662715285767248673788815