| L(s) = 1 | + 2·2-s + 4·4-s − 9·5-s + 7·7-s + 8·8-s − 18·10-s − 45·11-s − 16·13-s + 14·14-s + 16·16-s − 66·17-s + 11·19-s − 36·20-s − 90·22-s − 27·23-s − 44·25-s − 32·26-s + 28·28-s + 12·29-s − 169·31-s + 32·32-s − 132·34-s − 63·35-s + 209·37-s + 22·38-s − 72·40-s − 291·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.804·5-s + 0.377·7-s + 0.353·8-s − 0.569·10-s − 1.23·11-s − 0.341·13-s + 0.267·14-s + 1/4·16-s − 0.941·17-s + 0.132·19-s − 0.402·20-s − 0.872·22-s − 0.244·23-s − 0.351·25-s − 0.241·26-s + 0.188·28-s + 0.0768·29-s − 0.979·31-s + 0.176·32-s − 0.665·34-s − 0.304·35-s + 0.928·37-s + 0.0939·38-s − 0.284·40-s − 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 11 | \( 1 + 45 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 11 T + p^{3} T^{2} \) |
| 23 | \( 1 + 27 T + p^{3} T^{2} \) |
| 29 | \( 1 - 12 T + p^{3} T^{2} \) |
| 31 | \( 1 + 169 T + p^{3} T^{2} \) |
| 37 | \( 1 - 209 T + p^{3} T^{2} \) |
| 41 | \( 1 + 291 T + p^{3} T^{2} \) |
| 43 | \( 1 + 394 T + p^{3} T^{2} \) |
| 47 | \( 1 + 174 T + p^{3} T^{2} \) |
| 53 | \( 1 + 228 T + p^{3} T^{2} \) |
| 59 | \( 1 + 474 T + p^{3} T^{2} \) |
| 61 | \( 1 + 232 T + p^{3} T^{2} \) |
| 67 | \( 1 - 992 T + p^{3} T^{2} \) |
| 71 | \( 1 - 153 T + p^{3} T^{2} \) |
| 73 | \( 1 - 686 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1046 T + p^{3} T^{2} \) |
| 83 | \( 1 + 708 T + p^{3} T^{2} \) |
| 89 | \( 1 + 195 T + p^{3} T^{2} \) |
| 97 | \( 1 + 88 T + p^{3} T^{2} \) |
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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
−10.81410774943951625750654267572, −9.703496085810806096625449044112, −8.298067431070109976289268624717, −7.67974554065509359174984776589, −6.64445090174472980979861168868, −5.34159843486562465955769161832, −4.54362740845384691204283122423, −3.38414517790263229378933266447, −2.09866239075966204542304897544, 0,
2.09866239075966204542304897544, 3.38414517790263229378933266447, 4.54362740845384691204283122423, 5.34159843486562465955769161832, 6.64445090174472980979861168868, 7.67974554065509359174984776589, 8.298067431070109976289268624717, 9.703496085810806096625449044112, 10.81410774943951625750654267572
