| L(s) = 1 | − 5·3-s − 7·5-s − 26·7-s − 2·9-s − 11·11-s + 52·13-s + 35·15-s + 46·17-s − 96·19-s + 130·21-s + 27·23-s − 76·25-s + 145·27-s + 16·29-s − 293·31-s + 55·33-s + 182·35-s − 29·37-s − 260·39-s − 472·41-s − 110·43-s + 14·45-s − 224·47-s + 333·49-s − 230·51-s + 754·53-s + 77·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.626·5-s − 1.40·7-s − 0.0740·9-s − 0.301·11-s + 1.10·13-s + 0.602·15-s + 0.656·17-s − 1.15·19-s + 1.35·21-s + 0.244·23-s − 0.607·25-s + 1.03·27-s + 0.102·29-s − 1.69·31-s + 0.290·33-s + 0.878·35-s − 0.128·37-s − 1.06·39-s − 1.79·41-s − 0.390·43-s + 0.0463·45-s − 0.695·47-s + 0.970·49-s − 0.631·51-s + 1.95·53-s + 0.188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 + 96 T + p^{3} T^{2} \) |
| 23 | \( 1 - 27 T + p^{3} T^{2} \) |
| 29 | \( 1 - 16 T + p^{3} T^{2} \) |
| 31 | \( 1 + 293 T + p^{3} T^{2} \) |
| 37 | \( 1 + 29 T + p^{3} T^{2} \) |
| 41 | \( 1 + 472 T + p^{3} T^{2} \) |
| 43 | \( 1 + 110 T + p^{3} T^{2} \) |
| 47 | \( 1 + 224 T + p^{3} T^{2} \) |
| 53 | \( 1 - 754 T + p^{3} T^{2} \) |
| 59 | \( 1 - 825 T + p^{3} T^{2} \) |
| 61 | \( 1 + 548 T + p^{3} T^{2} \) |
| 67 | \( 1 + 123 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1001 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 79 | \( 1 - 526 T + p^{3} T^{2} \) |
| 83 | \( 1 + 158 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1217 T + p^{3} T^{2} \) |
| 97 | \( 1 + 263 T + p^{3} T^{2} \) |
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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
−15.11564239797692519708117432496, −13.37869599702103504396107526132, −12.39351030425872811801968553886, −11.28004623594834053748731134860, −10.19265231354656616959780975340, −8.576704438579042456112569388312, −6.79287207161281936191185534526, −5.66616212641346908893631684536, −3.60297344040032755627201064246, 0,
3.60297344040032755627201064246, 5.66616212641346908893631684536, 6.79287207161281936191185534526, 8.576704438579042456112569388312, 10.19265231354656616959780975340, 11.28004623594834053748731134860, 12.39351030425872811801968553886, 13.37869599702103504396107526132, 15.11564239797692519708117432496
