L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 12-s − 13-s − 2·14-s + 16-s + 18-s + 2·19-s + 2·21-s + 6·23-s − 24-s − 26-s − 27-s − 2·28-s + 8·31-s + 32-s + 36-s − 2·37-s + 2·38-s + 39-s + 6·41-s + 2·42-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 − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.436·21-s + 1.25·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 1.43·31-s + 0.176·32-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + 0.160·39-s + 0.937·41-s + 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.095728437\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.095728437\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.370296839521995809208788291669, −8.330848159112721404865557869779, −7.27707344485777707512281012317, −6.73478403534912402697136388658, −5.93286799828651801721907644062, −5.17686188643445276922627165113, −4.36656618875332976823977955578, −3.36468093384861562198182740569, −2.47628498116019091449481270494, −0.912909626419422780414937078348,
0.912909626419422780414937078348, 2.47628498116019091449481270494, 3.36468093384861562198182740569, 4.36656618875332976823977955578, 5.17686188643445276922627165113, 5.93286799828651801721907644062, 6.73478403534912402697136388658, 7.27707344485777707512281012317, 8.330848159112721404865557869779, 9.370296839521995809208788291669