| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 12-s + 2·13-s + 14-s + 16-s − 6·17-s + 18-s − 21-s − 23-s − 24-s − 5·25-s + 2·26-s − 27-s + 28-s − 6·29-s + 8·31-s + 32-s − 6·34-s + 36-s − 6·37-s − 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.986·37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−12.79966351505160, −12.29939427720914, −11.86942469182502, −11.38403360762818, −11.11130573504919, −10.74753880540200, −10.18340280377066, −9.654906024105053, −9.198175228068257, −8.604966781845583, −8.090711270759864, −7.720382847778957, −7.035643075757008, −6.682149119743207, −6.147195097484176, −5.860517760532325, −5.143409236302984, −4.890801597829165, −4.174103060089308, −3.928591790723961, −3.362536840766043, −2.462529961532356, −2.155229009062975, −1.513439306292871, −0.7919144789993771, 0,
0.7919144789993771, 1.513439306292871, 2.155229009062975, 2.462529961532356, 3.362536840766043, 3.928591790723961, 4.174103060089308, 4.890801597829165, 5.143409236302984, 5.860517760532325, 6.147195097484176, 6.682149119743207, 7.035643075757008, 7.720382847778957, 8.090711270759864, 8.604966781845583, 9.198175228068257, 9.654906024105053, 10.18340280377066, 10.74753880540200, 11.11130573504919, 11.38403360762818, 11.86942469182502, 12.29939427720914, 12.79966351505160
