| L(s) = 1 | − 2-s − 2.41·3-s + 4-s + 2.41·6-s − 2.41·7-s − 8-s + 2.82·9-s + 2·11-s − 2.41·12-s + 4.65·13-s + 2.41·14-s + 16-s − 17-s − 2.82·18-s − 4.82·19-s + 5.82·21-s − 2·22-s + 3.65·23-s + 2.41·24-s − 4.65·26-s + 0.414·27-s − 2.41·28-s − 4·29-s + 4.41·31-s − 32-s − 4.82·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.985·6-s − 0.912·7-s − 0.353·8-s + 0.942·9-s + 0.603·11-s − 0.696·12-s + 1.29·13-s + 0.645·14-s + 0.250·16-s − 0.242·17-s − 0.666·18-s − 1.10·19-s + 1.27·21-s − 0.426·22-s + 0.762·23-s + 0.492·24-s − 0.913·26-s + 0.0797·27-s − 0.456·28-s − 0.742·29-s + 0.792·31-s − 0.176·32-s − 0.840·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 + 0.171T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 4.75T + 71T^{2} \) |
| 73 | \( 1 - 9.65T + 73T^{2} \) |
| 79 | \( 1 + 7.24T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.901214652818954849941453451721, −9.007050805938735906069297396817, −8.200218290014752007679683564767, −6.71832043111226579719029833115, −6.51131517699287499092528042702, −5.68529021066380826611335734900, −4.42536424171613716591100755623, −3.20317866091861749638702633526, −1.40866691395920224115338189443, 0,
1.40866691395920224115338189443, 3.20317866091861749638702633526, 4.42536424171613716591100755623, 5.68529021066380826611335734900, 6.51131517699287499092528042702, 6.71832043111226579719029833115, 8.200218290014752007679683564767, 9.007050805938735906069297396817, 9.901214652818954849941453451721
