| L(s) = 1 | − 4·2-s + 16·4-s − 98·7-s − 64·8-s + 354·11-s − 404·13-s + 392·14-s + 256·16-s − 654·17-s + 1.79e3·19-s − 1.41e3·22-s + 1.08e3·23-s + 1.61e3·26-s − 1.56e3·28-s − 5.75e3·29-s + 1.01e4·31-s − 1.02e3·32-s + 2.61e3·34-s − 5.55e3·37-s − 7.18e3·38-s − 1.29e4·41-s + 8.96e3·43-s + 5.66e3·44-s − 4.32e3·46-s + 5.40e3·47-s − 7.20e3·49-s − 6.46e3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.882·11-s − 0.663·13-s + 0.534·14-s + 1/4·16-s − 0.548·17-s + 1.14·19-s − 0.623·22-s + 0.425·23-s + 0.468·26-s − 0.377·28-s − 1.27·29-s + 1.90·31-s − 0.176·32-s + 0.388·34-s − 0.666·37-s − 0.807·38-s − 1.20·41-s + 0.739·43-s + 0.441·44-s − 0.301·46-s + 0.356·47-s − 3/7·49-s − 0.331·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 p^{2} T + p^{5} T^{2} \) |
| 11 | \( 1 - 354 T + p^{5} T^{2} \) |
| 13 | \( 1 + 404 T + p^{5} T^{2} \) |
| 17 | \( 1 + 654 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1796 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1080 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5754 T + p^{5} T^{2} \) |
| 31 | \( 1 - 10196 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5552 T + p^{5} T^{2} \) |
| 41 | \( 1 + 12960 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8968 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5400 T + p^{5} T^{2} \) |
| 53 | \( 1 + 8214 T + p^{5} T^{2} \) |
| 59 | \( 1 - 3954 T + p^{5} T^{2} \) |
| 61 | \( 1 - 962 T + p^{5} T^{2} \) |
| 67 | \( 1 - 4 p^{2} T + p^{5} T^{2} \) |
| 71 | \( 1 + 56148 T + p^{5} T^{2} \) |
| 73 | \( 1 - 85690 T + p^{5} T^{2} \) |
| 79 | \( 1 + 26044 T + p^{5} T^{2} \) |
| 83 | \( 1 - 93468 T + p^{5} T^{2} \) |
| 89 | \( 1 - 73428 T + p^{5} T^{2} \) |
| 97 | \( 1 + 128978 T + p^{5} T^{2} \) |
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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
−9.615649915167507618280806017823, −9.198250096773299137676400329500, −8.057543025489506319340361467200, −7.03738876676990034967022060890, −6.38807980390302601683693385675, −5.13201452535480333887326197228, −3.72838528178439747365091634297, −2.63086052041865832980236588900, −1.25116522715184502107567295009, 0,
1.25116522715184502107567295009, 2.63086052041865832980236588900, 3.72838528178439747365091634297, 5.13201452535480333887326197228, 6.38807980390302601683693385675, 7.03738876676990034967022060890, 8.057543025489506319340361467200, 9.198250096773299137676400329500, 9.615649915167507618280806017823
