L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 11-s + 12-s + 14-s + 16-s − 18-s − 6·19-s − 21-s − 22-s − 2·23-s − 24-s + 27-s − 28-s + 2·29-s − 6·31-s − 32-s + 33-s + 36-s + 2·37-s + 6·38-s + 4·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 1.37·19-s − 0.218·21-s − 0.213·22-s − 0.417·23-s − 0.204·24-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 1.07·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s + 0.328·37-s + 0.973·38-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.69340304850577, −16.17773576925917, −15.65050485883875, −15.05714456960083, −14.50608332943880, −14.00557710579138, −13.26764745856404, −12.53571749019322, −12.35661416524565, −11.35117579920878, −10.79147664753359, −10.33697364762989, −9.527730512061872, −9.124694687090642, −8.626217441601604, −7.810926894954638, −7.471886512578690, −6.490674626009903, −6.223502331160387, −5.241101951203987, −4.219608103542244, −3.737417795986279, −2.718064996709682, −2.162370291381913, −1.188440936938369, 0,
1.188440936938369, 2.162370291381913, 2.718064996709682, 3.737417795986279, 4.219608103542244, 5.241101951203987, 6.223502331160387, 6.490674626009903, 7.471886512578690, 7.810926894954638, 8.626217441601604, 9.124694687090642, 9.527730512061872, 10.33697364762989, 10.79147664753359, 11.35117579920878, 12.35661416524565, 12.53571749019322, 13.26764745856404, 14.00557710579138, 14.50608332943880, 15.05714456960083, 15.65050485883875, 16.17773576925917, 16.69340304850577