| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 13-s + 15-s − 17-s + 19-s + 21-s + 23-s + 25-s − 27-s + 29-s + 31-s + 35-s − 37-s − 39-s − 41-s + 43-s − 45-s + 47-s + 49-s + 51-s − 53-s − 57-s − 59-s + 61-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 13-s + 15-s − 17-s + 19-s + 21-s + 23-s + 25-s − 27-s + 29-s + 31-s + 35-s − 37-s − 39-s − 41-s + 43-s − 45-s + 47-s + 49-s + 51-s − 53-s − 57-s − 59-s + 61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8128976988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8128976988\) |
| \(L(1)\) |
\(\approx\) |
\(0.6697898042\) |
| \(L(1)\) |
\(\approx\) |
\(0.6697898042\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.32005264367336474866091155509, −28.96596614905071554921367189104, −28.42118493953989373280113983020, −27.22885730288130602742019685781, −26.36852085333239331170521203497, −24.8644313677777209027372523428, −23.676953187510776830588572487266, −22.89033733022005848454303275810, −22.18034526869468808642779752034, −20.67238884142794973539594451137, −19.38837875854101961212519248292, −18.5057979300433088154887027967, −17.22257780305087694883660839119, −15.894777281237203004382324847772, −15.64183384858501162522590636844, −13.527894962370063199302277343555, −12.41353236887593202576114768172, −11.4243001095607364466285374091, −10.39277387022713974860724786817, −8.8650999009525606659970504683, −7.21400784145741596856969536140, −6.24628111273611913389858367498, −4.68364736988830086724666555597, −3.3521221386264899320004540654, −0.75461682057381672196926706822,
0.75461682057381672196926706822, 3.3521221386264899320004540654, 4.68364736988830086724666555597, 6.24628111273611913389858367498, 7.21400784145741596856969536140, 8.8650999009525606659970504683, 10.39277387022713974860724786817, 11.4243001095607364466285374091, 12.41353236887593202576114768172, 13.527894962370063199302277343555, 15.64183384858501162522590636844, 15.894777281237203004382324847772, 17.22257780305087694883660839119, 18.5057979300433088154887027967, 19.38837875854101961212519248292, 20.67238884142794973539594451137, 22.18034526869468808642779752034, 22.89033733022005848454303275810, 23.676953187510776830588572487266, 24.8644313677777209027372523428, 26.36852085333239331170521203497, 27.22885730288130602742019685781, 28.42118493953989373280113983020, 28.96596614905071554921367189104, 30.32005264367336474866091155509
