| L(s) = 1 | − 1.41·3-s − 0.999·9-s + 2·11-s + 2.82·13-s − 4.24·17-s + 4.24·19-s − 8·23-s − 5·25-s + 5.65·27-s − 6·29-s + 8.48·31-s − 2.82·33-s + 2·37-s − 4.00·39-s + 4.24·41-s + 6·43-s − 2.82·47-s + 6·51-s − 6·53-s − 6·57-s − 12.7·59-s − 5.65·61-s + 12·67-s + 11.3·69-s + 4·71-s + 1.41·73-s + 7.07·75-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.333·9-s + 0.603·11-s + 0.784·13-s − 1.02·17-s + 0.973·19-s − 1.66·23-s − 25-s + 1.08·27-s − 1.11·29-s + 1.52·31-s − 0.492·33-s + 0.328·37-s − 0.640·39-s + 0.662·41-s + 0.914·43-s − 0.412·47-s + 0.840·51-s − 0.824·53-s − 0.794·57-s − 1.65·59-s − 0.724·61-s + 1.46·67-s + 1.36·69-s + 0.474·71-s + 0.165·73-s + 0.816·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.217790480063225570257654950291, −7.64608067666295556878301618295, −6.43108591847073478429958042870, −6.17932381996412853777100953366, −5.39787288275064834184367370998, −4.40142134429809664105625059678, −3.70900613798035432805635948105, −2.52107878595946861721135283451, −1.33194651121093810617994551282, 0,
1.33194651121093810617994551282, 2.52107878595946861721135283451, 3.70900613798035432805635948105, 4.40142134429809664105625059678, 5.39787288275064834184367370998, 6.17932381996412853777100953366, 6.43108591847073478429958042870, 7.64608067666295556878301618295, 8.217790480063225570257654950291
