| L(s) = 1 | + 5-s − 5·11-s + 5·13-s + 17-s − 4·19-s − 6·23-s − 4·25-s − 2·31-s + 37-s + 2·41-s − 43-s + 6·47-s + 9·53-s − 5·55-s − 4·59-s − 6·61-s + 5·65-s + 67-s − 6·71-s + 7·73-s + 79-s − 83-s + 85-s + 7·89-s − 4·95-s + 3·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.50·11-s + 1.38·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s − 4/5·25-s − 0.359·31-s + 0.164·37-s + 0.312·41-s − 0.152·43-s + 0.875·47-s + 1.23·53-s − 0.674·55-s − 0.520·59-s − 0.768·61-s + 0.620·65-s + 0.122·67-s − 0.712·71-s + 0.819·73-s + 0.112·79-s − 0.109·83-s + 0.108·85-s + 0.741·89-s − 0.410·95-s + 0.304·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.54810655100813, −13.48634255427193, −12.95508914753298, −12.40711149232765, −11.93718684392254, −11.29199528519012, −10.81138832099297, −10.38129071095207, −10.10739124478802, −9.416133815982729, −8.840453651728759, −8.423260656857132, −7.803459344228786, −7.610532397345585, −6.743863671908655, −6.135241241267493, −5.818713512225328, −5.398084598020214, −4.637139505641390, −4.049375134529482, −3.561529216578130, −2.822814028884509, −2.169009009815109, −1.772831373553119, −0.8092165996891768, 0,
0.8092165996891768, 1.772831373553119, 2.169009009815109, 2.822814028884509, 3.561529216578130, 4.049375134529482, 4.637139505641390, 5.398084598020214, 5.818713512225328, 6.135241241267493, 6.743863671908655, 7.610532397345585, 7.803459344228786, 8.423260656857132, 8.840453651728759, 9.416133815982729, 10.10739124478802, 10.38129071095207, 10.81138832099297, 11.29199528519012, 11.93718684392254, 12.40711149232765, 12.95508914753298, 13.48634255427193, 13.54810655100813
