| L(s) = 1 | − 2.64·2-s + 8.72·3-s − 1.02·4-s − 19.5·5-s − 23.0·6-s + 7·7-s + 23.8·8-s + 49.1·9-s + 51.7·10-s + 41.9·11-s − 8.96·12-s + 72.2·13-s − 18.4·14-s − 170.·15-s − 54.7·16-s + 17·17-s − 129.·18-s + 30.1·19-s + 20.1·20-s + 61.0·21-s − 110.·22-s + 27.6·23-s + 208.·24-s + 258.·25-s − 190.·26-s + 193.·27-s − 7.18·28-s + ⋯ |
| L(s) = 1 | − 0.933·2-s + 1.67·3-s − 0.128·4-s − 1.75·5-s − 1.56·6-s + 0.377·7-s + 1.05·8-s + 1.82·9-s + 1.63·10-s + 1.14·11-s − 0.215·12-s + 1.54·13-s − 0.352·14-s − 2.94·15-s − 0.855·16-s + 0.242·17-s − 1.69·18-s + 0.364·19-s + 0.224·20-s + 0.634·21-s − 1.07·22-s + 0.250·23-s + 1.76·24-s + 2.06·25-s − 1.43·26-s + 1.37·27-s − 0.0485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.389885270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389885270\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 2.64T + 8T^{2} \) |
| 3 | \( 1 - 8.72T + 27T^{2} \) |
| 5 | \( 1 + 19.5T + 125T^{2} \) |
| 11 | \( 1 - 41.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.2T + 2.19e3T^{2} \) |
| 19 | \( 1 - 30.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 27.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 213.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 7.12T + 5.06e4T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 21.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 533.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 598.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 32.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 891.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 103.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 380.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 931.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 980.T + 9.12e5T^{2} \) |
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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
−13.25430430901559988009476896793, −11.82141443155719949404214909778, −10.78151041324581416071463204098, −9.294323946596640689255338277375, −8.629003316496288656568900735379, −8.007023108584281320454594535174, −7.17243816042502289239194358215, −4.20809667798585514751793488517, −3.54568411298690253109115832231, −1.23579772631249405156352991532,
1.23579772631249405156352991532, 3.54568411298690253109115832231, 4.20809667798585514751793488517, 7.17243816042502289239194358215, 8.007023108584281320454594535174, 8.629003316496288656568900735379, 9.294323946596640689255338277375, 10.78151041324581416071463204098, 11.82141443155719949404214909778, 13.25430430901559988009476896793
