L(s) = 1 | − 3-s − 2·4-s − 2·5-s − 7-s − 2·9-s + 2·12-s − 3·13-s + 2·15-s + 4·16-s + 4·17-s − 2·19-s + 4·20-s + 21-s − 2·23-s − 25-s + 5·27-s + 2·28-s − 2·31-s + 2·35-s + 4·36-s − 8·37-s + 3·39-s − 3·41-s + 3·43-s + 4·45-s + 10·47-s − 4·48-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s − 0.377·7-s − 2/3·9-s + 0.577·12-s − 0.832·13-s + 0.516·15-s + 16-s + 0.970·17-s − 0.458·19-s + 0.894·20-s + 0.218·21-s − 0.417·23-s − 1/5·25-s + 0.962·27-s + 0.377·28-s − 0.359·31-s + 0.338·35-s + 2/3·36-s − 1.31·37-s + 0.480·39-s − 0.468·41-s + 0.457·43-s + 0.596·45-s + 1.45·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 131 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.48457426181910954076901948751, −12.03123178923170132332703726387, −10.74854190018749957647436539215, −9.665115817523666883701028008312, −8.516334285594619945876123551799, −7.50814170283998456841073263207, −5.90193276607434244505102901463, −4.77305398609316631799123478484, −3.44179560367931521008003815658, 0,
3.44179560367931521008003815658, 4.77305398609316631799123478484, 5.90193276607434244505102901463, 7.50814170283998456841073263207, 8.516334285594619945876123551799, 9.665115817523666883701028008312, 10.74854190018749957647436539215, 12.03123178923170132332703726387, 12.48457426181910954076901948751