| L(s) = 1 | + 1.34·2-s − 0.184·4-s + 2.53·5-s + 7-s − 2.94·8-s + 3.41·10-s + 0.467·11-s + 5.82·13-s + 1.34·14-s − 3.59·16-s + 3.87·17-s − 2.18·19-s − 0.467·20-s + 0.630·22-s − 0.106·23-s + 1.41·25-s + 7.84·26-s − 0.184·28-s + 8.78·29-s − 7.68·31-s + 1.04·32-s + 5.22·34-s + 2.53·35-s − 7.68·37-s − 2.94·38-s − 7.45·40-s − 2.22·41-s + ⋯ |
| L(s) = 1 | + 0.952·2-s − 0.0923·4-s + 1.13·5-s + 0.377·7-s − 1.04·8-s + 1.07·10-s + 0.141·11-s + 1.61·13-s + 0.360·14-s − 0.899·16-s + 0.940·17-s − 0.501·19-s − 0.104·20-s + 0.134·22-s − 0.0221·23-s + 0.282·25-s + 1.53·26-s − 0.0349·28-s + 1.63·29-s − 1.37·31-s + 0.184·32-s + 0.896·34-s + 0.428·35-s − 1.26·37-s − 0.477·38-s − 1.17·40-s − 0.347·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.610574635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.610574635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 11 | \( 1 - 0.467T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 + 0.106T + 23T^{2} \) |
| 29 | \( 1 - 8.78T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 - 1.22T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 + 0.716T + 53T^{2} \) |
| 59 | \( 1 - 0.736T + 59T^{2} \) |
| 61 | \( 1 - 0.958T + 61T^{2} \) |
| 67 | \( 1 + 9.63T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 2.73T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−10.77463542736284382954414799722, −9.926324352541480838031922888434, −8.947907712055612490407147763954, −8.299237540438401501409117924631, −6.72357238737280526519684747320, −5.89571155133672503574805541277, −5.29820456314595406450409040327, −4.12364885605112318005770689367, −3.10347672796169061490169401766, −1.56246823708341917748907078270,
1.56246823708341917748907078270, 3.10347672796169061490169401766, 4.12364885605112318005770689367, 5.29820456314595406450409040327, 5.89571155133672503574805541277, 6.72357238737280526519684747320, 8.299237540438401501409117924631, 8.947907712055612490407147763954, 9.926324352541480838031922888434, 10.77463542736284382954414799722
