| L(s) = 1 | + 2·3-s − 7-s + 9-s + 2·11-s + 4·13-s − 2·17-s − 8·19-s − 2·21-s + 23-s − 4·27-s + 6·29-s + 4·33-s − 8·37-s + 8·39-s − 10·41-s + 10·43-s − 8·47-s + 49-s − 4·51-s − 12·53-s − 16·57-s − 6·59-s + 10·61-s − 63-s + 10·67-s + 2·69-s + 10·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.485·17-s − 1.83·19-s − 0.436·21-s + 0.208·23-s − 0.769·27-s + 1.11·29-s + 0.696·33-s − 1.31·37-s + 1.28·39-s − 1.56·41-s + 1.52·43-s − 1.16·47-s + 1/7·49-s − 0.560·51-s − 1.64·53-s − 2.11·57-s − 0.781·59-s + 1.28·61-s − 0.125·63-s + 1.22·67-s + 0.240·69-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.953869565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.953869565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.20722901562023, −13.90483550717042, −13.19108472511828, −12.90390228027625, −12.41029816675192, −11.65872684835102, −11.17040147742655, −10.64600525403061, −10.13429146245448, −9.485970172421319, −8.975039015476837, −8.484485922542642, −8.379841811073106, −7.646195735626962, −6.734723900374819, −6.555283704775028, −6.021104499651589, −5.132942012883383, −4.459812266822220, −3.842682434495195, −3.431448608888064, −2.823695248935168, −2.047634834604794, −1.612576241640163, −0.5155910775440802,
0.5155910775440802, 1.612576241640163, 2.047634834604794, 2.823695248935168, 3.431448608888064, 3.842682434495195, 4.459812266822220, 5.132942012883383, 6.021104499651589, 6.555283704775028, 6.734723900374819, 7.646195735626962, 8.379841811073106, 8.484485922542642, 8.975039015476837, 9.485970172421319, 10.13429146245448, 10.64600525403061, 11.17040147742655, 11.65872684835102, 12.41029816675192, 12.90390228027625, 13.19108472511828, 13.90483550717042, 14.20722901562023
