| L(s) = 1 | + 1.41·5-s + 7-s − 2·11-s − 2.82·13-s − 4.24·17-s + 19-s + 3.65·23-s − 2.99·25-s − 0.585·29-s − 4.82·31-s + 1.41·35-s + 10.4·37-s + 1.65·41-s + 2·43-s − 7.07·47-s + 49-s − 11.8·53-s − 2.82·55-s − 13.6·59-s + 0.828·61-s − 4.00·65-s − 4.48·67-s − 15.4·71-s − 11.6·73-s − 2·77-s + 9.17·79-s − 1.41·83-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + 0.377·7-s − 0.603·11-s − 0.784·13-s − 1.02·17-s + 0.229·19-s + 0.762·23-s − 0.599·25-s − 0.108·29-s − 0.867·31-s + 0.239·35-s + 1.72·37-s + 0.258·41-s + 0.304·43-s − 1.03·47-s + 0.142·49-s − 1.63·53-s − 0.381·55-s − 1.77·59-s + 0.106·61-s − 0.496·65-s − 0.547·67-s − 1.82·71-s − 1.36·73-s − 0.227·77-s + 1.03·79-s − 0.155·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 0.585T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.74842576303359474357358729524, −7.38459854021968739465688102888, −6.36071377448973138260891710138, −5.78156473253580039371533791354, −4.88558181924899562483121931182, −4.42293299928465566223611421464, −3.14298853307600965889940771986, −2.37745428103109298094846144212, −1.52830066126480666744683537254, 0,
1.52830066126480666744683537254, 2.37745428103109298094846144212, 3.14298853307600965889940771986, 4.42293299928465566223611421464, 4.88558181924899562483121931182, 5.78156473253580039371533791354, 6.36071377448973138260891710138, 7.38459854021968739465688102888, 7.74842576303359474357358729524
