| L(s) = 1 | − 5-s − 11-s − 2·13-s − 6·17-s + 8·19-s − 6·23-s + 25-s − 6·29-s + 2·31-s + 2·37-s − 8·43-s + 12·47-s − 6·53-s + 55-s − 6·59-s − 8·61-s + 2·65-s − 2·67-s + 10·73-s − 8·79-s − 12·83-s + 6·85-s + 6·89-s − 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 1.45·17-s + 1.83·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.328·37-s − 1.21·43-s + 1.75·47-s − 0.824·53-s + 0.134·55-s − 0.781·59-s − 1.02·61-s + 0.248·65-s − 0.244·67-s + 1.17·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s + 0.635·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2898009418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2898009418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.42045372089423, −11.89179972331057, −11.67815113373230, −11.07073167033801, −10.82919026352281, −10.06143955250216, −9.843108712158537, −9.252230885827842, −8.916623612426786, −8.319612276432558, −7.789383261658664, −7.458112249853706, −7.116562877242171, −6.426974911538845, −6.003459500295090, −5.447189202428992, −4.922578542627534, −4.518407775557730, −3.927042051716812, −3.477927559524689, −2.822722469815421, −2.373662148358164, −1.728046321414101, −1.089877228495971, −0.1435780879244050,
0.1435780879244050, 1.089877228495971, 1.728046321414101, 2.373662148358164, 2.822722469815421, 3.477927559524689, 3.927042051716812, 4.518407775557730, 4.922578542627534, 5.447189202428992, 6.003459500295090, 6.426974911538845, 7.116562877242171, 7.458112249853706, 7.789383261658664, 8.319612276432558, 8.916623612426786, 9.252230885827842, 9.843108712158537, 10.06143955250216, 10.82919026352281, 11.07073167033801, 11.67815113373230, 11.89179972331057, 12.42045372089423
