L(s) = 1 | − 2-s − 3.19·3-s + 4-s − 4.19·5-s + 3.19·6-s − 1.32·7-s − 8-s + 7.20·9-s + 4.19·10-s − 3.19·12-s + 3.52·13-s + 1.32·14-s + 13.4·15-s + 16-s + 2.32·17-s − 7.20·18-s − 19-s − 4.19·20-s + 4.22·21-s + 2.52·23-s + 3.19·24-s + 12.5·25-s − 3.52·26-s − 13.4·27-s − 1.32·28-s + 10.4·29-s − 13.4·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.84·3-s + 0.5·4-s − 1.87·5-s + 1.30·6-s − 0.499·7-s − 0.353·8-s + 2.40·9-s + 1.32·10-s − 0.922·12-s + 0.978·13-s + 0.353·14-s + 3.45·15-s + 0.250·16-s + 0.563·17-s − 1.69·18-s − 0.229·19-s − 0.937·20-s + 0.922·21-s + 0.527·23-s + 0.652·24-s + 2.51·25-s − 0.691·26-s − 2.58·27-s − 0.249·28-s + 1.93·29-s − 2.44·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3857979271\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3857979271\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 3.19T + 3T^{2} \) |
| 5 | \( 1 + 4.19T + 5T^{2} \) |
| 7 | \( 1 + 1.32T + 7T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 17 | \( 1 - 2.32T + 17T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 6.38T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 + 2.34T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 + 6.72T + 59T^{2} \) |
| 61 | \( 1 - 8.60T + 61T^{2} \) |
| 67 | \( 1 - 6.21T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.67T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 - 1.80T + 83T^{2} \) |
| 89 | \( 1 - 6.04T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.156587066035140155431260083801, −7.53545037666368258660113774195, −6.82761923396334985560532482446, −6.35483368341180859962164369208, −5.50383847510715975467682487653, −4.57769294656834178492125926709, −3.95354267075113859127260151226, −3.05119658116346577158174961752, −1.16892330791519608977018898783, −0.51610211686086639244445759792,
0.51610211686086639244445759792, 1.16892330791519608977018898783, 3.05119658116346577158174961752, 3.95354267075113859127260151226, 4.57769294656834178492125926709, 5.50383847510715975467682487653, 6.35483368341180859962164369208, 6.82761923396334985560532482446, 7.53545037666368258660113774195, 8.156587066035140155431260083801