| L(s) = 1 | − 4·2-s + 16·4-s + 49·7-s − 64·8-s − 243·9-s + 384·11-s − 236·13-s − 196·14-s + 256·16-s + 1.17e3·17-s + 972·18-s − 1.10e3·19-s − 1.53e3·22-s − 1.40e3·23-s + 944·26-s + 784·28-s − 3.85e3·29-s + 88·31-s − 1.02e3·32-s − 4.68e3·34-s − 3.88e3·36-s + 1.32e4·37-s + 4.40e3·38-s − 1.33e4·41-s + 2.50e3·43-s + 6.14e3·44-s + 5.60e3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s + 0.956·11-s − 0.387·13-s − 0.267·14-s + 1/4·16-s + 0.983·17-s + 0.707·18-s − 0.699·19-s − 0.676·22-s − 0.551·23-s + 0.273·26-s + 0.188·28-s − 0.850·29-s + 0.0164·31-s − 0.176·32-s − 0.695·34-s − 1/2·36-s + 1.58·37-s + 0.494·38-s − 1.23·41-s + 0.206·43-s + 0.478·44-s + 0.390·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + p^{5} T^{2} \) |
| 11 | \( 1 - 384 T + p^{5} T^{2} \) |
| 13 | \( 1 + 236 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1172 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1100 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1400 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3854 T + p^{5} T^{2} \) |
| 31 | \( 1 - 88 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13240 T + p^{5} T^{2} \) |
| 41 | \( 1 + 13338 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2504 T + p^{5} T^{2} \) |
| 47 | \( 1 - 14728 T + p^{5} T^{2} \) |
| 53 | \( 1 + 11232 T + p^{5} T^{2} \) |
| 59 | \( 1 - 652 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1494 T + p^{5} T^{2} \) |
| 67 | \( 1 + 18232 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28356 T + p^{5} T^{2} \) |
| 73 | \( 1 - 70892 T + p^{5} T^{2} \) |
| 79 | \( 1 + 79828 T + p^{5} T^{2} \) |
| 83 | \( 1 + 83712 T + p^{5} T^{2} \) |
| 89 | \( 1 + 93290 T + p^{5} T^{2} \) |
| 97 | \( 1 + 91068 T + p^{5} T^{2} \) |
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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
−10.09113840115463200734506559809, −9.222214712105934900346549584542, −8.381188995766349836685550516928, −7.55609472383329798312627815592, −6.38183831916474989392899099479, −5.47906948662687958838023176179, −3.99739579493469817528286286073, −2.66653907760061642876213468864, −1.38480895374814738063859204802, 0,
1.38480895374814738063859204802, 2.66653907760061642876213468864, 3.99739579493469817528286286073, 5.47906948662687958838023176179, 6.38183831916474989392899099479, 7.55609472383329798312627815592, 8.381188995766349836685550516928, 9.222214712105934900346549584542, 10.09113840115463200734506559809
