| L(s) = 1 | + 1.41·2-s − 1.41·3-s + 4.41·5-s − 2.00·6-s + 7-s − 2.82·8-s − 0.999·9-s + 6.24·10-s − 4.24·11-s − 13-s + 1.41·14-s − 6.24·15-s − 4.00·16-s − 1.41·17-s − 1.41·18-s + 1.24·19-s − 1.41·21-s − 6·22-s − 0.171·23-s + 4·24-s + 14.4·25-s − 1.41·26-s + 5.65·27-s + 5.82·29-s − 8.82·30-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.816·3-s + 1.97·5-s − 0.816·6-s + 0.377·7-s − 0.999·8-s − 0.333·9-s + 1.97·10-s − 1.27·11-s − 0.277·13-s + 0.377·14-s − 1.61·15-s − 1.00·16-s − 0.342·17-s − 0.333·18-s + 0.285·19-s − 0.308·21-s − 1.27·22-s − 0.0357·23-s + 0.816·24-s + 2.89·25-s − 0.277·26-s + 1.08·27-s + 1.08·29-s − 1.61·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.335659892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335659892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 4.41T + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 0.171T + 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 + 5.24T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 + 5.82T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + 0.757T + 73T^{2} \) |
| 79 | \( 1 + 1.48T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−13.91530704403553248137339920165, −13.18598525856379489294894562606, −12.31039705786425851174123193269, −10.92173925275991688173885509546, −9.956676790942336342764905946639, −8.720090509496169016660368964555, −6.55216153418865536823210553119, −5.46067764345373140398153217209, −5.09174108420603531238146202946, −2.59168801763164316544158957285,
2.59168801763164316544158957285, 5.09174108420603531238146202946, 5.46067764345373140398153217209, 6.55216153418865536823210553119, 8.720090509496169016660368964555, 9.956676790942336342764905946639, 10.92173925275991688173885509546, 12.31039705786425851174123193269, 13.18598525856379489294894562606, 13.91530704403553248137339920165
