| L(s) = 1 | − 3-s − 4·5-s − 4·7-s + 9-s + 13-s + 4·15-s + 6·17-s + 4·19-s + 4·21-s − 4·23-s + 11·25-s − 27-s + 6·29-s − 8·31-s + 16·35-s − 10·37-s − 39-s + 4·41-s − 4·43-s − 4·45-s + 6·47-s + 9·49-s − 6·51-s + 6·53-s − 4·57-s + 6·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 1.03·15-s + 1.45·17-s + 0.917·19-s + 0.872·21-s − 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 2.70·35-s − 1.64·37-s − 0.160·39-s + 0.624·41-s − 0.609·43-s − 0.596·45-s + 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s + 0.781·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.33328235690453, −13.82345915721640, −13.10965380797987, −12.64667119618997, −12.17879549709813, −11.91363271850495, −11.54967723605198, −10.72588445912713, −10.38458428221198, −9.881779806369088, −9.302832236749722, −8.643017942877430, −8.168302139329272, −7.480225953970934, −7.164539567095498, −6.731467925586629, −5.957160663332745, −5.488606631324951, −4.910631009328175, −3.936348962389765, −3.776020954586280, −3.256476103862080, −2.647450309405434, −1.363531769896483, −0.6465827656281006, 0,
0.6465827656281006, 1.363531769896483, 2.647450309405434, 3.256476103862080, 3.776020954586280, 3.936348962389765, 4.910631009328175, 5.488606631324951, 5.957160663332745, 6.731467925586629, 7.164539567095498, 7.480225953970934, 8.168302139329272, 8.643017942877430, 9.302832236749722, 9.881779806369088, 10.38458428221198, 10.72588445912713, 11.54967723605198, 11.91363271850495, 12.17879549709813, 12.64667119618997, 13.10965380797987, 13.82345915721640, 14.33328235690453
