| L(s) = 1 | + 1.47·2-s − 2.68·3-s + 0.189·4-s − 0.774·5-s − 3.97·6-s − 4.67·7-s − 2.67·8-s + 4.22·9-s − 1.14·10-s − 0.545·11-s − 0.508·12-s − 0.907·13-s − 6.92·14-s + 2.08·15-s − 4.34·16-s − 5.99·17-s + 6.24·18-s + 4.38·19-s − 0.146·20-s + 12.5·21-s − 0.806·22-s + 1.04·23-s + 7.20·24-s − 4.40·25-s − 1.34·26-s − 3.28·27-s − 0.884·28-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 1.55·3-s + 0.0946·4-s − 0.346·5-s − 1.62·6-s − 1.76·7-s − 0.947·8-s + 1.40·9-s − 0.362·10-s − 0.164·11-s − 0.146·12-s − 0.251·13-s − 1.84·14-s + 0.537·15-s − 1.08·16-s − 1.45·17-s + 1.47·18-s + 1.00·19-s − 0.0327·20-s + 2.74·21-s − 0.171·22-s + 0.218·23-s + 1.46·24-s − 0.880·25-s − 0.263·26-s − 0.632·27-s − 0.167·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4683107742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4683107742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 \) |
| good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 + 0.774T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 + 0.545T + 11T^{2} \) |
| 13 | \( 1 + 0.907T + 13T^{2} \) |
| 17 | \( 1 + 5.99T + 17T^{2} \) |
| 19 | \( 1 - 4.38T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + 7.25T + 37T^{2} \) |
| 43 | \( 1 - 6.22T + 43T^{2} \) |
| 47 | \( 1 - 3.97T + 47T^{2} \) |
| 53 | \( 1 - 5.35T + 53T^{2} \) |
| 59 | \( 1 - 8.87T + 59T^{2} \) |
| 61 | \( 1 - 1.38T + 61T^{2} \) |
| 67 | \( 1 - 5.63T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 + 5.61T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 1.95T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−9.550899054290538909451785941077, −8.739310858078160327224140792829, −7.14907888426822574158371583585, −6.71727990923817121107734675679, −5.86461757528180872272075014780, −5.43088412711919951450863391094, −4.41329765765747869890401929098, −3.72029070399467225098705197802, −2.67359331885479268820224153955, −0.41104067902700400817973492749,
0.41104067902700400817973492749, 2.67359331885479268820224153955, 3.72029070399467225098705197802, 4.41329765765747869890401929098, 5.43088412711919951450863391094, 5.86461757528180872272075014780, 6.71727990923817121107734675679, 7.14907888426822574158371583585, 8.739310858078160327224140792829, 9.550899054290538909451785941077
