L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 2·11-s − 2·13-s + 14-s + 16-s + 2·17-s − 4·19-s + 2·22-s − 23-s + 2·26-s − 28-s − 8·29-s + 2·31-s − 32-s − 2·34-s − 4·37-s + 4·38-s − 2·41-s + 8·43-s − 2·44-s + 46-s + 12·47-s + 49-s − 2·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.426·22-s − 0.208·23-s + 0.392·26-s − 0.188·28-s − 1.48·29-s + 0.359·31-s − 0.176·32-s − 0.342·34-s − 0.657·37-s + 0.648·38-s − 0.312·41-s + 1.21·43-s − 0.301·44-s + 0.147·46-s + 1.75·47-s + 1/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−14.40878170241767, −13.84347364156769, −13.32391847136367, −12.73072879926328, −12.33360668605962, −11.90353164335959, −11.21867402111800, −10.65089935233217, −10.37206563918271, −9.833420442524894, −9.190557926338249, −8.917298593967909, −8.208452597578762, −7.631818306986336, −7.356131355048365, −6.651208156650070, −6.139800542988928, −5.439756686922243, −5.127581738406975, −4.011910087446845, −3.815880309542261, −2.731041629323965, −2.434606927864632, −1.679596287637530, −0.7355730172935726, 0,
0.7355730172935726, 1.679596287637530, 2.434606927864632, 2.731041629323965, 3.815880309542261, 4.011910087446845, 5.127581738406975, 5.439756686922243, 6.139800542988928, 6.651208156650070, 7.356131355048365, 7.631818306986336, 8.208452597578762, 8.917298593967909, 9.190557926338249, 9.833420442524894, 10.37206563918271, 10.65089935233217, 11.21867402111800, 11.90353164335959, 12.33360668605962, 12.73072879926328, 13.32391847136367, 13.84347364156769, 14.40878170241767