| L(s) = 1 | + 2-s − 0.734·3-s + 4-s − 5-s − 0.734·6-s + 0.223·7-s + 8-s − 2.46·9-s − 10-s + 5.68·11-s − 0.734·12-s + 6.23·13-s + 0.223·14-s + 0.734·15-s + 16-s − 5.95·17-s − 2.46·18-s − 5.38·19-s − 20-s − 0.163·21-s + 5.68·22-s + 7.72·23-s − 0.734·24-s + 25-s + 6.23·26-s + 4.01·27-s + 0.223·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.424·3-s + 0.5·4-s − 0.447·5-s − 0.299·6-s + 0.0843·7-s + 0.353·8-s − 0.820·9-s − 0.316·10-s + 1.71·11-s − 0.212·12-s + 1.72·13-s + 0.0596·14-s + 0.189·15-s + 0.250·16-s − 1.44·17-s − 0.579·18-s − 1.23·19-s − 0.223·20-s − 0.0357·21-s + 1.21·22-s + 1.61·23-s − 0.149·24-s + 0.200·25-s + 1.22·26-s + 0.771·27-s + 0.0421·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.753883616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.753883616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 0.734T + 3T^{2} \) |
| 7 | \( 1 - 0.223T + 7T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 5.95T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 - 7.72T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 - 8.22T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 - 1.01T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 - 5.26T + 59T^{2} \) |
| 61 | \( 1 - 1.52T + 61T^{2} \) |
| 67 | \( 1 + 3.64T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 6.98T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.376640600862685157221841982725, −6.89007855553639350121822150110, −6.45622500428881421705906932863, −6.26208532816402988183517356510, −5.03127726158657367107309000872, −4.49134622294937116614302076473, −3.69729864403280912900835657629, −3.07951166084849616858673298442, −1.84439172294530514240660944336, −0.828574998911987433648303975392,
0.828574998911987433648303975392, 1.84439172294530514240660944336, 3.07951166084849616858673298442, 3.69729864403280912900835657629, 4.49134622294937116614302076473, 5.03127726158657367107309000872, 6.26208532816402988183517356510, 6.45622500428881421705906932863, 6.89007855553639350121822150110, 8.376640600862685157221841982725
