| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 12-s − 2·13-s + 4·14-s + 16-s − 17-s − 18-s + 4·21-s − 8·23-s + 24-s + 2·26-s − 27-s − 4·28-s − 4·29-s + 2·31-s − 32-s + 34-s + 36-s + 2·39-s + 8·41-s − 4·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.872·21-s − 1.66·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.755·28-s − 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s + 0.320·39-s + 1.24·41-s − 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.88248342510533, −12.32356013739242, −11.96148861668290, −11.60732515394273, −10.88291802619368, −10.53480947970713, −10.15971272237641, −9.590217023812869, −9.409817809541791, −8.926430143476070, −8.269457868046456, −7.689078909618626, −7.303217071632867, −6.915295491464691, −6.216567468656626, −5.965569214582498, −5.688474502979230, −4.786944220992989, −4.224034883561935, −3.785015375941687, −3.095440412681920, −2.575632428677949, −2.050156192189270, −1.294644451836393, −0.4812283116220898, 0,
0.4812283116220898, 1.294644451836393, 2.050156192189270, 2.575632428677949, 3.095440412681920, 3.785015375941687, 4.224034883561935, 4.786944220992989, 5.688474502979230, 5.965569214582498, 6.216567468656626, 6.915295491464691, 7.303217071632867, 7.689078909618626, 8.269457868046456, 8.926430143476070, 9.409817809541791, 9.590217023812869, 10.15971272237641, 10.53480947970713, 10.88291802619368, 11.60732515394273, 11.96148861668290, 12.32356013739242, 12.88248342510533
