| L(s) = 1 | + 2·2-s + 5.50·3-s + 4·4-s − 5·5-s + 11.0·6-s + 0.0500·7-s + 8·8-s + 3.26·9-s − 10·10-s + 35.1·11-s + 22.0·12-s + 86.1·13-s + 0.100·14-s − 27.5·15-s + 16·16-s + 8.82·17-s + 6.53·18-s + 106.·19-s − 20·20-s + 0.275·21-s + 70.3·22-s + 23·23-s + 44.0·24-s + 25·25-s + 172.·26-s − 130.·27-s + 0.200·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.05·3-s + 0.5·4-s − 0.447·5-s + 0.748·6-s + 0.00270·7-s + 0.353·8-s + 0.120·9-s − 0.316·10-s + 0.964·11-s + 0.529·12-s + 1.83·13-s + 0.00191·14-s − 0.473·15-s + 0.250·16-s + 0.125·17-s + 0.0855·18-s + 1.28·19-s − 0.223·20-s + 0.00286·21-s + 0.681·22-s + 0.208·23-s + 0.374·24-s + 0.200·25-s + 1.29·26-s − 0.930·27-s + 0.00135·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.844251851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.844251851\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 5.50T + 27T^{2} \) |
| 7 | \( 1 - 0.0500T + 343T^{2} \) |
| 11 | \( 1 - 35.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.82T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 280.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 93.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 58.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 508.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 407.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 129.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 299.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 596.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 692.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 842.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 409.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−11.62191604736660636454882356058, −11.21710880206054870212065473521, −9.630199338544330494958673907239, −8.717211303196833470815703803826, −7.84289999927998245593390808055, −6.68167572917697175433594143500, −5.47209119178626565991916149599, −3.79221102266587888045541856029, −3.34954089972216128614177621414, −1.56068778982998718654644880591,
1.56068778982998718654644880591, 3.34954089972216128614177621414, 3.79221102266587888045541856029, 5.47209119178626565991916149599, 6.68167572917697175433594143500, 7.84289999927998245593390808055, 8.717211303196833470815703803826, 9.630199338544330494958673907239, 11.21710880206054870212065473521, 11.62191604736660636454882356058
