L(s) = 1 | + 3-s − 3·7-s + 9-s + 11-s + 3·17-s + 3·19-s − 3·21-s + 3·23-s + 27-s − 6·29-s − 10·31-s + 33-s − 7·37-s + 7·41-s + 4·43-s + 9·47-s + 2·49-s + 3·51-s + 12·53-s + 3·57-s + 7·59-s − 4·61-s − 3·63-s + 12·67-s + 3·69-s − 15·71-s + 4·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.727·17-s + 0.688·19-s − 0.654·21-s + 0.625·23-s + 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.174·33-s − 1.15·37-s + 1.09·41-s + 0.609·43-s + 1.31·47-s + 2/7·49-s + 0.420·51-s + 1.64·53-s + 0.397·57-s + 0.911·59-s − 0.512·61-s − 0.377·63-s + 1.46·67-s + 0.361·69-s − 1.78·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.160241119\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.160241119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.913427490469652327755003368825, −7.20736896984304336568773519007, −6.86012164028745123631659095101, −5.72473444421808677376625262421, −5.39629903669030407303213887088, −4.06622801591281928207540078696, −3.58079973002407376867585330908, −2.88133901138195777634933795194, −1.91845895565519043884763382872, −0.73275497102316179637953621795,
0.73275497102316179637953621795, 1.91845895565519043884763382872, 2.88133901138195777634933795194, 3.58079973002407376867585330908, 4.06622801591281928207540078696, 5.39629903669030407303213887088, 5.72473444421808677376625262421, 6.86012164028745123631659095101, 7.20736896984304336568773519007, 7.913427490469652327755003368825