| L(s) = 1 | − 2-s + 3.20·3-s + 4-s + 3.48·5-s − 3.20·6-s − 8-s + 7.28·9-s − 3.48·10-s + 1.72·11-s + 3.20·12-s + 2.57·13-s + 11.1·15-s + 16-s − 5.59·17-s − 7.28·18-s + 3.54·19-s + 3.48·20-s − 1.72·22-s + 6.68·23-s − 3.20·24-s + 7.12·25-s − 2.57·26-s + 13.7·27-s + 3.34·29-s − 11.1·30-s − 6.25·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.85·3-s + 0.5·4-s + 1.55·5-s − 1.30·6-s − 0.353·8-s + 2.42·9-s − 1.10·10-s + 0.520·11-s + 0.925·12-s + 0.713·13-s + 2.88·15-s + 0.250·16-s − 1.35·17-s − 1.71·18-s + 0.813·19-s + 0.778·20-s − 0.367·22-s + 1.39·23-s − 0.654·24-s + 1.42·25-s − 0.504·26-s + 2.64·27-s + 0.620·29-s − 2.03·30-s − 1.12·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.644418613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.644418613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 - 3.20T + 3T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 + 5.59T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 + 6.25T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 0.0484T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 7.90T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 59 | \( 1 + 0.691T + 59T^{2} \) |
| 61 | \( 1 + 9.00T + 61T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 8.93T + 73T^{2} \) |
| 79 | \( 1 + 9.70T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 7.30T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.470041260468337261206582058216, −7.17438759482805519057861206525, −7.00590885212690243966563265767, −6.12317925320796838602876267772, −5.17893207996508561228594460599, −4.17159220744392385339312525679, −3.19998737440940161905006484551, −2.65299585772299008181395317194, −1.76026764784172905843751140309, −1.33499457393108344779047583595,
1.33499457393108344779047583595, 1.76026764784172905843751140309, 2.65299585772299008181395317194, 3.19998737440940161905006484551, 4.17159220744392385339312525679, 5.17893207996508561228594460599, 6.12317925320796838602876267772, 7.00590885212690243966563265767, 7.17438759482805519057861206525, 8.470041260468337261206582058216
