| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s − 12-s + 2·13-s + 2·14-s + 16-s + 17-s + 18-s + 4·19-s − 2·21-s − 6·23-s − 24-s − 5·25-s + 2·26-s − 27-s + 2·28-s + 10·31-s + 32-s + 34-s + 36-s − 8·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s − 1.25·23-s − 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s + 0.377·28-s + 1.79·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s − 1.31·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.97283627420730, −13.67792194721176, −13.42514806531645, −12.52415918844272, −12.06659579181514, −11.78936367786689, −11.43606309003443, −10.71055874073095, −10.38046063160774, −9.799207340026177, −9.288549253989097, −8.418505982253087, −7.958709242810429, −7.672082509404632, −6.853098569414936, −6.331290744319985, −5.935508571406058, −5.318424213749316, −4.804019766757604, −4.411585515981826, −3.572482526664436, −3.271855598828181, −2.248607081469341, −1.672099679651000, −1.077633403647618, 0,
1.077633403647618, 1.672099679651000, 2.248607081469341, 3.271855598828181, 3.572482526664436, 4.411585515981826, 4.804019766757604, 5.318424213749316, 5.935508571406058, 6.331290744319985, 6.853098569414936, 7.672082509404632, 7.958709242810429, 8.418505982253087, 9.288549253989097, 9.799207340026177, 10.38046063160774, 10.71055874073095, 11.43606309003443, 11.78936367786689, 12.06659579181514, 12.52415918844272, 13.42514806531645, 13.67792194721176, 13.97283627420730
