| L(s) = 1 | − 2·4-s − 3·5-s − 7-s − 11-s + 4·16-s + 2·17-s + 6·20-s − 4·23-s + 4·25-s + 2·28-s + 5·31-s + 3·35-s − 10·37-s − 5·41-s + 4·43-s + 2·44-s + 3·47-s + 49-s − 5·53-s + 3·55-s + 10·59-s − 5·61-s − 8·64-s + 10·67-s − 4·68-s − 15·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 4-s − 1.34·5-s − 0.377·7-s − 0.301·11-s + 16-s + 0.485·17-s + 1.34·20-s − 0.834·23-s + 4/5·25-s + 0.377·28-s + 0.898·31-s + 0.507·35-s − 1.64·37-s − 0.780·41-s + 0.609·43-s + 0.301·44-s + 0.437·47-s + 1/7·49-s − 0.686·53-s + 0.404·55-s + 1.30·59-s − 0.640·61-s − 64-s + 1.22·67-s − 0.485·68-s − 1.78·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−12.99970064220394, −12.54578889847307, −12.17144576776916, −11.88845000709680, −11.27838687061132, −10.75180054374640, −10.22127122102000, −9.853089802700701, −9.430469996567746, −8.637558708785526, −8.453867937529044, −8.051565729336767, −7.498339248229348, −7.080362643720166, −6.506333429574852, −5.737746108359007, −5.456395456128902, −4.690878953854299, −4.414675491879332, −3.758906068089590, −3.470132502228454, −2.943196860009019, −2.113123205690921, −1.257702869216546, −0.5436350788349484, 0,
0.5436350788349484, 1.257702869216546, 2.113123205690921, 2.943196860009019, 3.470132502228454, 3.758906068089590, 4.414675491879332, 4.690878953854299, 5.456395456128902, 5.737746108359007, 6.506333429574852, 7.080362643720166, 7.498339248229348, 8.051565729336767, 8.453867937529044, 8.637558708785526, 9.430469996567746, 9.853089802700701, 10.22127122102000, 10.75180054374640, 11.27838687061132, 11.88845000709680, 12.17144576776916, 12.54578889847307, 12.99970064220394
