L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 2·11-s − 12-s + 2·13-s + 14-s + 16-s + 8·17-s + 18-s − 2·19-s − 21-s − 2·22-s − 24-s + 2·26-s − 27-s + 28-s − 6·29-s + 6·31-s + 32-s + 2·33-s + 8·34-s + 36-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 − 0.603·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.235·18-s − 0.458·19-s − 0.218·21-s − 0.426·22-s − 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.348·33-s + 1.37·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.254991124\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.254991124\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.21059932252477432620307698625, −9.176896752088126288360451561822, −7.888629906615445707051997160584, −7.49923731991253040447983361802, −6.14142726842051751076884063333, −5.70369798620661172413606267539, −4.74594274055821180416788620663, −3.82103395693658892210659718139, −2.65182798269852143160411026186, −1.17226570083808247348509158767,
1.17226570083808247348509158767, 2.65182798269852143160411026186, 3.82103395693658892210659718139, 4.74594274055821180416788620663, 5.70369798620661172413606267539, 6.14142726842051751076884063333, 7.49923731991253040447983361802, 7.888629906615445707051997160584, 9.176896752088126288360451561822, 10.21059932252477432620307698625