| L(s) = 1 | − 2·5-s + 4·11-s + 2·13-s + 17-s + 4·19-s + 8·23-s − 25-s − 6·29-s − 2·37-s + 10·41-s + 4·43-s − 6·53-s − 8·55-s + 4·59-s − 6·61-s − 4·65-s + 12·67-s − 8·71-s + 6·73-s + 12·83-s − 2·85-s − 6·89-s − 8·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + 0.554·13-s + 0.242·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 1.11·29-s − 0.328·37-s + 1.56·41-s + 0.609·43-s − 0.824·53-s − 1.07·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 1.46·67-s − 0.949·71-s + 0.702·73-s + 1.31·83-s − 0.216·85-s − 0.635·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−13.83141019604519, −13.30661710479139, −12.78760983840639, −12.31730617806268, −11.80513813243911, −11.42388146470440, −10.95865362841609, −10.68246193571923, −9.683801691664808, −9.366154334918154, −9.020917789658040, −8.407337010987903, −7.695086382098962, −7.548278604066816, −6.812243193693464, −6.460619699321464, −5.690275121219452, −5.276265605407799, −4.556427199478243, −3.931137178765810, −3.645794835287465, −3.047790684406258, −2.311479033828819, −1.277442215123560, −1.043943476530579, 0,
1.043943476530579, 1.277442215123560, 2.311479033828819, 3.047790684406258, 3.645794835287465, 3.931137178765810, 4.556427199478243, 5.276265605407799, 5.690275121219452, 6.460619699321464, 6.812243193693464, 7.548278604066816, 7.695086382098962, 8.407337010987903, 9.020917789658040, 9.366154334918154, 9.683801691664808, 10.68246193571923, 10.95865362841609, 11.42388146470440, 11.80513813243911, 12.31730617806268, 12.78760983840639, 13.30661710479139, 13.83141019604519
