L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 2.82·11-s − 0.828·13-s − 15-s + 7.65·17-s + 2.82·19-s + 21-s − 4·23-s + 25-s − 27-s + 7.65·29-s − 2.82·31-s − 2.82·33-s − 35-s + 11.6·37-s + 0.828·39-s − 7.65·41-s − 1.65·43-s + 45-s + 11.3·47-s + 49-s − 7.65·51-s − 10.4·53-s + 2.82·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.852·11-s − 0.229·13-s − 0.258·15-s + 1.85·17-s + 0.648·19-s + 0.218·21-s − 0.834·23-s + 0.200·25-s − 0.192·27-s + 1.42·29-s − 0.508·31-s − 0.492·33-s − 0.169·35-s + 1.91·37-s + 0.132·39-s − 1.19·41-s − 0.252·43-s + 0.149·45-s + 1.65·47-s + 0.142·49-s − 1.07·51-s − 1.44·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.989823649\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.989823649\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 7.17T + 97T^{2} \) |
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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
−7.76632973087251468703746996747, −7.34566889859702100275225136517, −6.26305195421136963550807856051, −6.08154994631803165407991308029, −5.22437721043821979999081570554, −4.47513907850799098350507130633, −3.56310356959109460826159096102, −2.83294676071737423937573133352, −1.60128779532713125992427644199, −0.800054793163270154658604901274,
0.800054793163270154658604901274, 1.60128779532713125992427644199, 2.83294676071737423937573133352, 3.56310356959109460826159096102, 4.47513907850799098350507130633, 5.22437721043821979999081570554, 6.08154994631803165407991308029, 6.26305195421136963550807856051, 7.34566889859702100275225136517, 7.76632973087251468703746996747