| L(s) = 1 | + 3-s + 9-s − 2·11-s − 5·13-s + 2·17-s − 3·19-s + 2·23-s + 27-s + 8·29-s + 31-s − 2·33-s + 5·37-s − 5·39-s + 2·41-s + 7·43-s − 8·47-s + 2·51-s + 2·53-s − 3·57-s − 10·59-s − 2·61-s − 11·67-s + 2·69-s − 12·71-s + 3·73-s − 17·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.38·13-s + 0.485·17-s − 0.688·19-s + 0.417·23-s + 0.192·27-s + 1.48·29-s + 0.179·31-s − 0.348·33-s + 0.821·37-s − 0.800·39-s + 0.312·41-s + 1.06·43-s − 1.16·47-s + 0.280·51-s + 0.274·53-s − 0.397·57-s − 1.30·59-s − 0.256·61-s − 1.34·67-s + 0.240·69-s − 1.42·71-s + 0.351·73-s − 1.91·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.01028373551702, −13.17487174679973, −12.95756056947035, −12.47679099025371, −11.91538900168431, −11.55224105874205, −10.71560431688569, −10.39149426807382, −9.926853928794081, −9.486832835397687, −8.851091950558332, −8.466039498011977, −7.828513235564255, −7.429533414710022, −7.080359597829324, −6.215205688902527, −5.916319986604757, −5.055544167785768, −4.552198753605474, −4.337665181550602, −3.241915034186109, −2.905690775370455, −2.401853788354369, −1.702659329567102, −0.8662938049924055, 0,
0.8662938049924055, 1.702659329567102, 2.401853788354369, 2.905690775370455, 3.241915034186109, 4.337665181550602, 4.552198753605474, 5.055544167785768, 5.916319986604757, 6.215205688902527, 7.080359597829324, 7.429533414710022, 7.828513235564255, 8.466039498011977, 8.851091950558332, 9.486832835397687, 9.926853928794081, 10.39149426807382, 10.71560431688569, 11.55224105874205, 11.91538900168431, 12.47679099025371, 12.95756056947035, 13.17487174679973, 14.01028373551702
