L(s) = 1 | + (0.988 − 0.149i)2-s + (−0.123 − 0.0841i)3-s + (0.955 − 0.294i)4-s + (0.0747 − 0.997i)5-s + (−0.134 − 0.0648i)6-s + (−0.955 + 0.294i)7-s + (0.900 − 0.433i)8-s + (−0.357 − 0.910i)9-s + (−0.0747 − 0.997i)10-s + (−0.142 − 0.0440i)12-s + (−0.900 + 0.433i)14-s + (−0.0931 + 0.116i)15-s + (0.826 − 0.563i)16-s + (−0.488 − 0.846i)18-s + (−0.222 − 0.974i)20-s + (0.142 + 0.0440i)21-s + ⋯ |
L(s) = 1 | + (0.988 − 0.149i)2-s + (−0.123 − 0.0841i)3-s + (0.955 − 0.294i)4-s + (0.0747 − 0.997i)5-s + (−0.134 − 0.0648i)6-s + (−0.955 + 0.294i)7-s + (0.900 − 0.433i)8-s + (−0.357 − 0.910i)9-s + (−0.0747 − 0.997i)10-s + (−0.142 − 0.0440i)12-s + (−0.900 + 0.433i)14-s + (−0.0931 + 0.116i)15-s + (0.826 − 0.563i)16-s + (−0.488 − 0.846i)18-s + (−0.222 − 0.974i)20-s + (0.142 + 0.0440i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.637880624\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.637880624\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.988 + 0.149i)T \) |
| 5 | \( 1 + (-0.0747 + 0.997i)T \) |
| 7 | \( 1 + (0.955 - 0.294i)T \) |
good | 3 | \( 1 + (0.123 + 0.0841i)T + (0.365 + 0.930i)T^{2} \) |
| 11 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.21 - 1.12i)T + (0.0747 + 0.997i)T^{2} \) |
| 29 | \( 1 + (-0.326 - 1.42i)T + (-0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 41 | \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.658 - 0.317i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (1.78 - 0.268i)T + (0.955 - 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (-1.57 - 0.487i)T + (0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.19 - 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.535 + 1.36i)T + (-0.733 + 0.680i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.988888070273113292603457088585, −9.335348482818127477478500314756, −8.521640225710505942702553294203, −7.21421884977349681689858286282, −6.45581776671646651140850114284, −5.62175240573906578466880996779, −4.92278826984063234757459586040, −3.69901585624529768332314194852, −2.96214243559143450847195649451, −1.32979940979002654998114676492,
2.34149732217882607813321235871, 3.06767121299619250349847987051, 4.09586522516207137260005160658, 5.15670945580722490946346151406, 6.14183057057635373281404170216, 6.75251888769262539501614468444, 7.50899349538949276486467854601, 8.475145507598943549543526628745, 9.897045355510140777079089859806, 10.47676167486162811653869677816