| L(s) = 1 | − 3·3-s − 8.27·5-s + 7·7-s + 9·9-s − 72.0·11-s − 89.4·13-s + 24.8·15-s − 38.0·17-s + 46.9·19-s − 21·21-s − 65.1·23-s − 56.4·25-s − 27·27-s + 118.·29-s − 175.·31-s + 216.·33-s − 57.9·35-s − 157.·37-s + 268.·39-s + 293.·41-s − 230.·43-s − 74.5·45-s − 139.·47-s + 49·49-s + 114.·51-s + 128.·53-s + 596.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.740·5-s + 0.377·7-s + 0.333·9-s − 1.97·11-s − 1.90·13-s + 0.427·15-s − 0.543·17-s + 0.566·19-s − 0.218·21-s − 0.590·23-s − 0.451·25-s − 0.192·27-s + 0.757·29-s − 1.01·31-s + 1.14·33-s − 0.279·35-s − 0.699·37-s + 1.10·39-s + 1.11·41-s − 0.819·43-s − 0.246·45-s − 0.432·47-s + 0.142·49-s + 0.313·51-s + 0.332·53-s + 1.46·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2718140061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2718140061\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 8.27T + 125T^{2} \) |
| 11 | \( 1 + 72.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 89.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 38.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 65.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 118.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 157.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 293.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 230.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 128.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 58.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 129.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 221.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 700.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 791.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 854.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 921.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.56e3T + 9.12e5T^{2} \) |
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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
−9.359316072851769232198493450326, −8.118850014182878643020843661794, −7.63163956407498061379112636400, −7.03520231072557564711633614782, −5.68020407355251566642528021679, −5.02405277841662530527076626532, −4.37388072437792487790328252330, −2.98827386049188764259619307761, −2.05263355326379615545968833042, −0.25089271005070443125425440372,
0.25089271005070443125425440372, 2.05263355326379615545968833042, 2.98827386049188764259619307761, 4.37388072437792487790328252330, 5.02405277841662530527076626532, 5.68020407355251566642528021679, 7.03520231072557564711633614782, 7.63163956407498061379112636400, 8.118850014182878643020843661794, 9.359316072851769232198493450326
