| L(s) = 1 | − 1.56·3-s − 0.0338·7-s − 0.550·9-s − 5.79·11-s + 3.11·13-s + 4.09·17-s + 0.00975·19-s + 0.0530·21-s − 2.32·23-s + 5.55·27-s + 5.42·29-s + 4.69·31-s + 9.07·33-s − 10.6·37-s − 4.88·39-s + 5.88·41-s − 0.480·43-s − 11.3·47-s − 6.99·49-s − 6.40·51-s + 6.98·53-s − 0.0152·57-s + 7.09·59-s − 5.31·61-s + 0.0186·63-s + 2.91·67-s + 3.63·69-s + ⋯ |
| L(s) = 1 | − 0.903·3-s − 0.0128·7-s − 0.183·9-s − 1.74·11-s + 0.865·13-s + 0.992·17-s + 0.00223·19-s + 0.0115·21-s − 0.484·23-s + 1.06·27-s + 1.00·29-s + 0.843·31-s + 1.57·33-s − 1.75·37-s − 0.781·39-s + 0.918·41-s − 0.0732·43-s − 1.65·47-s − 0.999·49-s − 0.896·51-s + 0.959·53-s − 0.00202·57-s + 0.923·59-s − 0.680·61-s + 0.00235·63-s + 0.356·67-s + 0.437·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8879791157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8879791157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 + 0.0338T + 7T^{2} \) |
| 11 | \( 1 + 5.79T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 - 4.09T + 17T^{2} \) |
| 19 | \( 1 - 0.00975T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 5.88T + 41T^{2} \) |
| 43 | \( 1 + 0.480T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 + 5.31T + 61T^{2} \) |
| 67 | \( 1 - 2.91T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 7.17T + 73T^{2} \) |
| 79 | \( 1 - 0.358T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 2.14T + 97T^{2} \) |
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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
−7.70352573991668218982049539349, −6.86133182506739236125616640753, −6.15616728105268106535816906765, −5.57795629396026374104606081498, −5.11887262246317663352861962372, −4.37589717402198738455847275008, −3.25425187224258087925180950590, −2.75604947447532208179239460469, −1.54740817363838131779022206198, −0.47397214447986362050194597610,
0.47397214447986362050194597610, 1.54740817363838131779022206198, 2.75604947447532208179239460469, 3.25425187224258087925180950590, 4.37589717402198738455847275008, 5.11887262246317663352861962372, 5.57795629396026374104606081498, 6.15616728105268106535816906765, 6.86133182506739236125616640753, 7.70352573991668218982049539349
