| L(s) = 1 | − 1.41·2-s + 1.41·3-s + 1.58·5-s − 2.00·6-s + 7-s + 2.82·8-s − 0.999·9-s − 2.24·10-s + 4.24·11-s − 13-s − 1.41·14-s + 2.24·15-s − 4.00·16-s + 1.41·17-s + 1.41·18-s − 7.24·19-s + 1.41·21-s − 6·22-s − 5.82·23-s + 4·24-s − 2.48·25-s + 1.41·26-s − 5.65·27-s + 0.171·29-s − 3.17·30-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.816·3-s + 0.709·5-s − 0.816·6-s + 0.377·7-s + 0.999·8-s − 0.333·9-s − 0.709·10-s + 1.27·11-s − 0.277·13-s − 0.377·14-s + 0.579·15-s − 1.00·16-s + 0.342·17-s + 0.333·18-s − 1.66·19-s + 0.308·21-s − 1.27·22-s − 1.21·23-s + 0.816·24-s − 0.497·25-s + 0.277·26-s − 1.08·27-s + 0.0318·29-s − 0.579·30-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
| \(L(1)\) |
\(\approx\) |
\(0.8035041043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8035041043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 - 0.171T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 0.171T + 53T^{2} \) |
| 59 | \( 1 - 0.343T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 1.58T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−14.19259351354014001633935862578, −13.32961004431274155578181993754, −11.81034674227379139892165090587, −10.43811267915924677562981769248, −9.453975216742547325282993143478, −8.710892224506815951873187739116, −7.78956849606875471561960327198, −6.15614991745813169223168872051, −4.20245225198679567841491642802, −1.98957828776268363334345914285,
1.98957828776268363334345914285, 4.20245225198679567841491642802, 6.15614991745813169223168872051, 7.78956849606875471561960327198, 8.710892224506815951873187739116, 9.453975216742547325282993143478, 10.43811267915924677562981769248, 11.81034674227379139892165090587, 13.32961004431274155578181993754, 14.19259351354014001633935862578
