| L(s) = 1 | − 1.65·2-s + 0.726·4-s + 0.377·7-s + 2.10·8-s + 1.37·11-s − 2.82·13-s − 0.622·14-s − 4.92·16-s + 6.37·17-s + 19-s − 2.27·22-s + 6.19·23-s + 4.65·26-s + 0.273·28-s + 3.37·29-s + 2.48·31-s + 3.92·32-s − 10.5·34-s − 5.58·37-s − 1.65·38-s − 8.50·41-s + 12.1·43-s + 1.00·44-s − 10.2·46-s − 6.87·47-s − 6.85·49-s − 2.04·52-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.363·4-s + 0.142·7-s + 0.743·8-s + 0.415·11-s − 0.782·13-s − 0.166·14-s − 1.23·16-s + 1.54·17-s + 0.229·19-s − 0.484·22-s + 1.29·23-s + 0.913·26-s + 0.0517·28-s + 0.627·29-s + 0.445·31-s + 0.693·32-s − 1.80·34-s − 0.917·37-s − 0.267·38-s − 1.32·41-s + 1.85·43-s + 0.150·44-s − 1.50·46-s − 1.00·47-s − 0.979·49-s − 0.284·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.027423111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027423111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 7 | \( 1 - 0.377T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 6.87T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 + 3.22T + 67T^{2} \) |
| 71 | \( 1 - 2.30T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 - 18.2T + 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 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
−8.437288030841224754423606032919, −7.76279201052805983447907356403, −7.20299626456469796822449563957, −6.48708038096055355562751741547, −5.30362904113739979880436818796, −4.81257645616887778638140445208, −3.71316498053751411295420668696, −2.75530088610732733393894131303, −1.57022497673754590660678124160, −0.73767620475483239462312260767,
0.73767620475483239462312260767, 1.57022497673754590660678124160, 2.75530088610732733393894131303, 3.71316498053751411295420668696, 4.81257645616887778638140445208, 5.30362904113739979880436818796, 6.48708038096055355562751741547, 7.20299626456469796822449563957, 7.76279201052805983447907356403, 8.437288030841224754423606032919
