L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 3·11-s − 12-s + 6·13-s − 2·14-s + 16-s + 2·17-s − 18-s − 19-s − 2·21-s + 3·22-s + 23-s + 24-s − 6·26-s − 27-s + 2·28-s − 5·29-s + 7·31-s − 32-s + 3·33-s − 2·34-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 − 0.904·11-s − 0.288·12-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.436·21-s + 0.639·22-s + 0.208·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s + 0.377·28-s − 0.928·29-s + 1.25·31-s − 0.176·32-s + 0.522·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.206246773\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.206246773\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.757200212616892175265832739646, −7.917299457438728173962997409673, −7.59459687533897826331384481870, −6.39314113168345366826329082740, −5.89305002560950434038581374398, −5.02732304208680071295811199334, −4.08303945028190895204768365550, −2.97335582172833890339437647975, −1.76837705298710235362000162386, −0.811950925460365122151880326718,
0.811950925460365122151880326718, 1.76837705298710235362000162386, 2.97335582172833890339437647975, 4.08303945028190895204768365550, 5.02732304208680071295811199334, 5.89305002560950434038581374398, 6.39314113168345366826329082740, 7.59459687533897826331384481870, 7.917299457438728173962997409673, 8.757200212616892175265832739646